Vector products: Difference between revisions
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def cross_product(other : Vector) |
def cross_product(other : Vector) |
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Vector.new(self.y * other.z - self.z * other.y, |
Vector.new(self.y * other.z - self.z * other.y, |
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self.z * other.x - self.x * other.z, |
self.z * other.x - self.x * other.z, |
||
self.x * other.y - self.y * other.x) |
self.x * other.y - self.y * other.x) |
||
end |
end |
||
Revision as of 19:25, 27 April 2020
You are encouraged to solve this task according to the task description, using any language you may know.
A vector is defined as having three dimensions as being represented by an ordered collection of three numbers: (X, Y, Z).
If you imagine a graph with the x and y axis being at right angles to each other and having a third, z axis coming out of the page, then a triplet of numbers, (X, Y, Z) would represent a point in the region, and a vector from the origin to the point.
Given the vectors:
A = (a1, a2, a3)
B = (b1, b2, b3)
C = (c1, c2, c3)
then the following common vector products are defined:
- The dot product (a scalar quantity)
- A • B = a1b1 + a2b2 + a3b3
- The cross product (a vector quantity)
- A x B = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)
- The scalar triple product (a scalar quantity)
- A • (B x C)
- The vector triple product (a vector quantity)
- A x (B x C)
- Task
Given the three vectors:
a = ( 3, 4, 5)
b = ( 4, 3, 5)
c = (-5, -12, -13)
- Create a named function/subroutine/method to compute the dot product of two vectors.
- Create a function to compute the cross product of two vectors.
- Optionally create a function to compute the scalar triple product of three vectors.
- Optionally create a function to compute the vector triple product of three vectors.
- Compute and display:
a • b - Compute and display:
a x b - Compute and display:
a • (b x c), the scalar triple product. - Compute and display:
a x (b x c), the vector triple product.
- References
- A starting page on Wolfram MathWorld is Vector Multiplication .
- Wikipedia dot product.
- Wikipedia cross product.
- Wikipedia triple product.
- Related tasks
Ada
not using Ada.Numerics.Real_Arrays, to show some features of the language.
Ada determines which function to call not only on the types of the parameters, but also on the return type. That way we can use the same name for all multiplications (scalar and cross). But, if we add another one to stretch the vector, we get an ambiguity error, since the compiler can't know if A*(B*C) with result-type Vector is meant to be A stretched by the scalar product of B and C, or the cross product of A and the result of the cross product of B and C. Here, I used type qualification to tell the compiler that the result of (B*C) is of type Vector.
vector.adb: <lang Ada>with Ada.Text_IO;
procedure Vector is
type Float_Vector is array (Positive range <>) of Float; package Float_IO is new Ada.Text_IO.Float_IO (Float);
procedure Vector_Put (X : Float_Vector) is
begin
Ada.Text_IO.Put ("(");
for I in X'Range loop
Float_IO.Put (X (I), Aft => 1, Exp => 0);
if I /= X'Last then
Ada.Text_IO.Put (", ");
end if;
end loop;
Ada.Text_IO.Put (")");
end Vector_Put;
-- cross product
function "*" (Left, Right : Float_Vector) return Float_Vector is
begin
if Left'Length /= Right'Length then
raise Constraint_Error with "vectors of different size in dot product";
end if;
if Left'Length /= 3 then
raise Constraint_Error with "dot product only implemented for R**3";
end if;
return Float_Vector'(Left (Left'First + 1) * Right (Right'First + 2) -
Left (Left'First + 2) * Right (Right'First + 1),
Left (Left'First + 2) * Right (Right'First) -
Left (Left'First) * Right (Right'First + 2),
Left (Left'First) * Right (Right'First + 1) -
Left (Left'First + 1) * Right (Right'First));
end "*";
-- scalar product
function "*" (Left, Right : Float_Vector) return Float is
Result : Float := 0.0;
I, J : Positive;
begin
if Left'Length /= Right'Length then
raise Constraint_Error with "vectors of different size in scalar product";
end if;
I := Left'First; J := Right'First;
while I <= Left'Last and then J <= Right'Last loop
Result := Result + Left (I) * Right (J);
I := I + 1; J := J + 1;
end loop;
return Result;
end "*";
-- stretching
function "*" (Left : Float_Vector; Right : Float) return Float_Vector is
Result : Float_Vector (Left'Range);
begin
for I in Left'Range loop
Result (I) := Left (I) * Right;
end loop;
return Result;
end "*";
A : constant Float_Vector := (3.0, 4.0, 5.0); B : constant Float_Vector := (4.0, 3.0, 5.0); C : constant Float_Vector := (-5.0, -12.0, -13.0);
begin
Ada.Text_IO.Put ("A: "); Vector_Put (A); Ada.Text_IO.New_Line;
Ada.Text_IO.Put ("B: "); Vector_Put (B); Ada.Text_IO.New_Line;
Ada.Text_IO.Put ("C: "); Vector_Put (C); Ada.Text_IO.New_Line;
Ada.Text_IO.New_Line;
Ada.Text_IO.Put ("A dot B = "); Float_IO.Put (A * B, Aft => 1, Exp => 0);
Ada.Text_IO.New_Line;
Ada.Text_IO.Put ("A x B = "); Vector_Put (A * B);
Ada.Text_IO.New_Line;
Ada.Text_IO.Put ("A dot (B x C) = "); Float_IO.Put (A * (B * C), Aft => 1, Exp => 0);
Ada.Text_IO.New_Line;
Ada.Text_IO.Put ("A x (B x C) = "); Vector_Put (A * Float_Vector'(B * C));
Ada.Text_IO.New_Line;
end Vector;</lang> Output:
A: ( 3.0, 4.0, 5.0) B: ( 4.0, 3.0, 5.0) C: (-5.0, -12.0, -13.0) A dot B = 49.0 A x B = ( 5.0, 5.0, -7.0) A dot (B x C) = 6.0 A x (B x C) = (-267.0, 204.0, -3.0)
ALGOL 68
Note: This specimen retains the original Python coding style.
<lang algol68>MODE FIELD = INT; FORMAT field fmt = $g(-0)$;
MODE VEC = [3]FIELD; FORMAT vec fmt = $"("f(field fmt)", "f(field fmt)", "f(field fmt)")"$;
PROC crossp = (VEC a, b)VEC:(
#Cross product of two 3D vectors# CO ASSERT(LWB a = LWB b AND UPB a = UPB b AND UPB b = 3 # "For 3D vectors only" #); CO (a[2]*b[3] - a[3]*b[2], a[3]*b[1] - a[1]*b[3], a[1]*b[2] - a[2]*b[1])
);
PRIO MAXLWB = 8, MINUPB=8;
OP MAXLWB = (VEC a, b)INT: (LWB a<LWB b|LWB a|LWB b); OP MINUPB = (VEC a, b)INT: (UPB a>UPB b|UPB a|UPB b);
PROC dotp = (VEC a, b)FIELD:(
#Dot product of two vectors# FIELD sum := 0; FOR i FROM a MAXLWB b TO a MINUPB b DO sum +:= a[i]*b[i] OD; sum
);
PROC scalartriplep = (VEC a, b, c)VEC:(
#Scalar triple product of three vectors: "a . (b x c)"# dotp(a, crossp(b, c))
);
PROC vectortriplep = (VEC a, b, c)VEC:(
#Vector triple product of three vectors: "a x (b x c)"# crossp(a, crossp(b, c))
);
- Declare some useful operators #
PRIO DOT = 5, X = 5; OP (VEC, VEC)FIELD DOT = dotp; OP (VEC, VEC)VEC X = crossp;
main:(
VEC a=(3, 4, 5), b=(4, 3, 5), c=(-5, -12, -13); printf(($"a = "f(vec fmt)"; b = "f(vec fmt)"; c = "f(vec fmt)l$ , a, b, c)); printf($"Using PROCedures:"l$); printf(($"a . b = "f(field fmt)l$, dotp(a,b))); printf(($"a x b = "f(vec fmt)l$, crossp(a,b))); printf(($"a . (b x c) = "f(field fmt)l$, scalartriplep(a, b, c))); printf(($"a x (b x c) = "f(vec fmt)l$, vectortriplep(a, b, c))); printf($"Using OPerators:"l$); printf(($"a . b = "f(field fmt)l$, a DOT b)); printf(($"a x b = "f(vec fmt)l$, a X b)); printf(($"a . (b x c) = "f(field fmt)l$, a DOT (b X c))); printf(($"a x (b x c) = "f(vec fmt)l$, a X (b X c)))
)</lang> Output:
a = (3, 4, 5); b = (4, 3, 5); c = (-5, -12, -13) Using PROCedures: a . b = 49 a x b = (5, 5, -7) a . (b x c) = 6 a x (b x c) = (-267, 204, -3) Using OPerators: a . b = 49 a x b = (5, 5, -7) a . (b x c) = 6 a x (b x c) = (-267, 204, -3)
ALGOL W
<lang algolw>begin
% define the Vector record type % record Vector( integer X, Y, Z );
% calculates the dot product of two Vectors %
integer procedure dotProduct( reference(Vector) value A, B ) ;
( X(A) * X(B) ) + ( Y(A) * Y(B) ) + ( Z(A) * Z(B) );
% calculates the cross product or two Vectors %
reference(Vector) procedure crossProduct( reference(Vector) value A, B ) ;
Vector( ( Y(A) * Z(B) ) - ( Z(A) * Y(B) )
, ( Z(A) * X(B) ) - ( X(A) * Z(B) )
, ( X(A) * Y(B) ) - ( Y(A) * X(B) )
);
% calculates the scaler triple product of two vectors %
integer procedure scalerTripleProduct( reference(Vector) value A, B, C ) ;
dotProduct( A, crossProduct( B, C ) );
% calculates the vector triple product of two vectors %
reference(Vector) procedure vectorTripleProduct( reference(Vector) value A, B, C ) ;
crossProduct( A, crossProduct( B, C ) );
% test the Vector routines %
begin
procedure writeonVector( reference(Vector) value v ) ;
writeon( "(", X(v), ", ", Y(v), ", ", Z(v), ")" );
Reference(Vector) a, b, c;
a := Vector( 3, 4, 5 );
b := Vector( 4, 3, 5 );
c := Vector( -5, -12, -13 );
i_w := 1; s_w := 0; % set output formatting %
write( " a: " ); writeonVector( a );
write( " b: " ); writeonVector( b );
write( " c: " ); writeonVector( c );
write( " a . b: ", dotProduct( a, b ) );
write( " a x b: " ); writeonVector( crossProduct( a, b ) );
write( "a . ( b x c ): ", scalerTripleProduct( a, b, c ) );
write( "a x ( b x c ): " ); writeonVector( vectorTripleProduct( a, b, c ) )
end
end.</lang>
- Output:
a: (3, 4, 5)
b: (4, 3, 5)
c: (-5, -12, -13)
a . b: 49
a x b: (5, 5, -7)
a . ( b x c ): 6
a x ( b x c ): (-267, 204, -3)
AutoHotkey
<lang AutoHotkey>V := {a: [3, 4, 5], b: [4, 3, 5], c: [-5, -12, -13]}
for key, val in V Out .= key " = (" val[1] ", " val[2] ", " val[3] ")`n"
CP := CrossProduct(V.a, V.b) VTP := VectorTripleProduct(V.a, V.b, V.c)
MsgBox, % Out "`na • b = " DotProduct(V.a, V.b) "`n" . "a x b = (" CP[1] ", " CP[2] ", " CP[3] ")`n" . "a • b x c = " ScalerTripleProduct(V.a, V.b, V.c) "`n" . "a x b x c = (" VTP[1] ", " VTP[2] ", " VTP[3] ")"
DotProduct(v1, v2) { return, v1[1] * v2[1] + v1[2] * v2[2] + v1[3] * v2[3] }
CrossProduct(v1, v2) { return, [v1[2] * v2[3] - v1[3] * v2[2] , v1[3] * v2[1] - v1[1] * v2[3] , v1[1] * v2[2] - v1[2] * v2[1]] }
ScalerTripleProduct(v1, v2, v3) { return, DotProduct(v1, CrossProduct(v2, v3)) }
VectorTripleProduct(v1, v2, v3) { return, CrossProduct(v1, CrossProduct(v2, v3)) }</lang> Output:
a = (3, 4, 5) b = (4, 3, 5) c = (-5, -12, -13) a • b = 49 a x b = (5, 5, -7) a • b x c = 6 a x b x c = (-267, 204, -3)
AWK
<lang awk>#!/usr/bin/awk -f BEGIN {
a[1] = 3; a[2]= 4; a[3] = 5;
b[1] = 4; b[2]= 3; b[3] = 5;
c[1] = -5; c[2]= -12; c[3] = -13;
print "a = ",printVec(a);
print "b = ",printVec(b);
print "c = ",printVec(c);
print "a.b = ",dot(a,b);
## upper case variables are used as temporary or intermediate results
cross(a,b,D);print "a.b = ",printVec(D);
cross(b,c,D);print "a.(b x c) = ",dot(a,D);
cross(b,c,D);cross(a,D,E); print "a x (b x c) = ",printVec(E);
}
function dot(A,B) {
return A[1]*B[1]+A[2]*B[2]+A[3]*B[3];
}
function cross(A,B,C) {
C[1] = A[2]*B[3]-A[3]*B[2];
C[2] = A[3]*B[1]-A[1]*B[3];
C[3] = A[1]*B[2]-A[2]*B[1];
}
function printVec(C) {
return "[ "C[1]" "C[2]" "C[3]" ]";
}</lang> Output:
a = [ 3 4 5 ] b = [ 4 3 5 ] c = [ -5 -12 -13 ] A.b = 49 a.b = [ 5 5 -7 ] a.(b x c) = 6 a x (b x c) = [ -267 204 -3 ]
BASIC256
<lang basic256>
a={3,4,5}:b={4,3,5}:c={-5,-12,-13}
print "A.B = "+dot_product(ref(a),ref(b)) call cross_product(ref(a),ref(b),ref(y)) Print "AxB = ("+y[0]+","+y[1]+","+y[2]+")" print "A.(BxC) = "+s_tri(ref(a),ref(b),ref(c)) call v_tri(ref(a),ref(b),ref(c),ref(x),ref(y)) Print "A x (BxC) = ("+y[0]+","+y[1]+","+y[2]+")"
function dot_product(ref(x1),ref(x2))
dot_product= 0
for t = 0 to 2
dot_product += x1[t]*x2[t]
next t
end function
subroutine cross_product(ref(x1),ref(x2),ref(y1))
y1={0,0,0}
y1[0]=x1[1]*x2[2]-x1[2]*x2[1]
y1[1]=x1[2]*x2[0]-x1[0]*x2[2]
y1[2]=x1[0]*x2[1]-x1[1]*x2[0]
end subroutine
function s_tri(ref(x1),ref(x2),ref(x3))
call cross_product(ref(x2),ref(x3),ref(y1)) s_tri=dot_product(ref(x1),ref(y1))
end function
subroutine v_tri(ref(x1),ref(x2),ref(x3),ref(y1),ref(y2))
call cross_product(ref(x2),ref(x3),ref(y1)) call cross_product(ref(x1),ref(y1),ref(y2))
end subroutine
</lang>
Output:
A.B = 49 AxB = (5,5,-7) A.(BxC) = 6 A x (BxC) = (-267,204,-3)
BBC BASIC
<lang bbcbasic> DIM a(2), b(2), c(2), d(2)
a() = 3, 4, 5
b() = 4, 3, 5
c() = -5, -12, -13
PRINT "a . b = "; FNdot(a(),b())
PROCcross(a(),b(),d())
PRINT "a x b = (";d(0)", ";d(1)", ";d(2)")"
PRINT "a . (b x c) = "; FNscalartriple(a(),b(),c())
PROCvectortriple(a(),b(),c(),d())
PRINT "a x (b x c) = (";d(0)", ";d(1)", ";d(2)")"
END
DEF FNdot(A(),B())
LOCAL C() : DIM C(0,0)
C() = A().B()
= C(0,0)
DEF PROCcross(A(),B(),C())
C() = A(1)*B(2)-A(2)*B(1), A(2)*B(0)-A(0)*B(2), A(0)*B(1)-A(1)*B(0)
ENDPROC
DEF FNscalartriple(A(),B(),C())
LOCAL D() : DIM D(2)
PROCcross(B(),C(),D())
= FNdot(A(),D())
DEF PROCvectortriple(A(),B(),C(),D())
PROCcross(B(),C(),D())
PROCcross(A(),D(),D())
ENDPROC</lang>
Output:
a . b = 49 a x b = (5, 5, -7) a . (b x c) = 6 a x (b x c) = (-267, 204, -3)
C
<lang c>#include<stdio.h>
typedef struct{ float i,j,k; }Vector;
Vector a = {3, 4, 5},b = {4, 3, 5},c = {-5, -12, -13};
float dotProduct(Vector a, Vector b) { return a.i*b.i+a.j*b.j+a.k*b.k; }
Vector crossProduct(Vector a,Vector b) { Vector c = {a.j*b.k - a.k*b.j, a.k*b.i - a.i*b.k, a.i*b.j - a.j*b.i};
return c; }
float scalarTripleProduct(Vector a,Vector b,Vector c) { return dotProduct(a,crossProduct(b,c)); }
Vector vectorTripleProduct(Vector a,Vector b,Vector c) { return crossProduct(a,crossProduct(b,c)); }
void printVector(Vector a) { printf("( %f, %f, %f)",a.i,a.j,a.k); }
int main() { printf("\n a = "); printVector(a); printf("\n b = "); printVector(b); printf("\n c = "); printVector(c); printf("\n a . b = %f",dotProduct(a,b)); printf("\n a x b = "); printVector(crossProduct(a,b)); printf("\n a . (b x c) = %f",scalarTripleProduct(a,b,c)); printf("\n a x (b x c) = "); printVector(vectorTripleProduct(a,b,c));
return 0; }</lang> Output:
a = ( 3.000000, 4.000000, 5.000000) b = ( 4.000000, 3.000000, 5.000000) c = ( -5.000000, -12.000000, -13.000000) a . b = 49.000000 a x b = ( 5.000000, 5.000000, -7.000000) a . (b x c) = 6.000000 a x (b x c) = ( -267.000000, 204.000000, -3.000000)
C#
<lang csharp>using System; using System.Windows.Media.Media3D;
class VectorProducts {
static double ScalarTripleProduct(Vector3D a, Vector3D b, Vector3D c)
{
return Vector3D.DotProduct(a, Vector3D.CrossProduct(b, c));
}
static Vector3D VectorTripleProduct(Vector3D a, Vector3D b, Vector3D c)
{
return Vector3D.CrossProduct(a, Vector3D.CrossProduct(b, c));
}
static void Main()
{
var a = new Vector3D(3, 4, 5);
var b = new Vector3D(4, 3, 5);
var c = new Vector3D(-5, -12, -13);
Console.WriteLine(Vector3D.DotProduct(a, b));
Console.WriteLine(Vector3D.CrossProduct(a, b));
Console.WriteLine(ScalarTripleProduct(a, b, c));
Console.WriteLine(VectorTripleProduct(a, b, c));
}
}</lang> Output:
49 5;5;-7 6 -267;204;-3
C++
<lang cpp>#include <iostream>
template< class T > class D3Vector {
template< class U > friend std::ostream & operator<<( std::ostream & , const D3Vector & ) ;
public :
D3Vector( T a , T b , T c ) {
x = a ;
y = b ;
z = c ;
}
T dotproduct ( const D3Vector & rhs ) {
T scalar = x * rhs.x + y * rhs.y + z * rhs.z ;
return scalar ;
}
D3Vector crossproduct ( const D3Vector & rhs ) {
T a = y * rhs.z - z * rhs.y ;
T b = z * rhs.x - x * rhs.z ;
T c = x * rhs.y - y * rhs.x ;
D3Vector product( a , b , c ) ;
return product ;
}
D3Vector triplevec( D3Vector & a , D3Vector & b ) {
return crossproduct ( a.crossproduct( b ) ) ;
}
T triplescal( D3Vector & a, D3Vector & b ) {
return dotproduct( a.crossproduct( b ) ) ;
}
private :
T x , y , z ;
} ;
template< class T > std::ostream & operator<< ( std::ostream & os , const D3Vector<T> & vec ) {
os << "( " << vec.x << " , " << vec.y << " , " << vec.z << " )" ; return os ;
}
int main( ) {
D3Vector<int> a( 3 , 4 , 5 ) , b ( 4 , 3 , 5 ) , c( -5 , -12 , -13 ) ; std::cout << "a . b : " << a.dotproduct( b ) << "\n" ; std::cout << "a x b : " << a.crossproduct( b ) << "\n" ; std::cout << "a . b x c : " << a.triplescal( b , c ) << "\n" ; std::cout << "a x b x c : " << a.triplevec( b , c ) << "\n" ; return 0 ;
}</lang>
Output:
a . b : 49 a x b : ( 5 , 5 , -7 ) a . b x c : 6 a x b x c : ( -267 , 204 , -3 )
Ceylon
<lang ceylon>shared void run() {
alias Vector => Float[3];
function dot(Vector a, Vector b) => a[0] * b[0] + a[1] * b[1] + a[2] * b[2];
function cross(Vector a, Vector b) => [ a[1] * b[2] - a[2] * b[1], a[2] * b[0] - a[0] * b[2], a[0] * b[1] - a[1] * b[0] ];
function scalarTriple(Vector a, Vector b, Vector c) => dot(a, cross(b, c));
function vectorTriple(Vector a, Vector b, Vector c) => cross(a, cross(b, c));
value a = [ 3.0, 4.0, 5.0 ]; value b = [ 4.0, 3.0, 5.0 ]; value c = [-5.0, -12.0, -13.0 ];
print("``a`` . ``b`` = ``dot(a, b)``"); print("``a`` X ``b`` = ``cross(a, b)``"); print("``a`` . ``b`` X ``c`` = ``scalarTriple(a, b, c)``"); print("``a`` X ``b`` X ``c`` = ``vectorTriple(a, b, c)``"); }</lang>
- Output:
[3.0, 4.0, 5.0] . [4.0, 3.0, 5.0] = 49.0 [3.0, 4.0, 5.0] X [4.0, 3.0, 5.0] = [5.0, 5.0, -7.0] [3.0, 4.0, 5.0] . [4.0, 3.0, 5.0] X [-5.0, -12.0, -13.0] = 6.0 [3.0, 4.0, 5.0] X [4.0, 3.0, 5.0] X [-5.0, -12.0, -13.0] = [-267.0, 204.0, -3.0]
Clojure
<lang clojure>(defrecord Vector [x y z])
(defn dot
[U V]
(+ (* (:x U) (:x V))
(* (:y U) (:y V))
(* (:z U) (:z V))))
(defn cross
[U V]
(new Vector
(- (* (:y U) (:z V)) (* (:z U) (:y V)))
(- (* (:z U) (:x V)) (* (:x U) (:z V)))
(- (* (:x U) (:y V)) (* (:y U) (:x V)))))
(let [a (new Vector 3 4 5)
b (new Vector 4 3 5)
c (new Vector -5 -12 -13)]
(doseq
[prod (list
(dot a b)
(cross a b)
(dot a (cross b c))
(cross a (cross b c)))]
(println prod)))</lang>
Output:
49
#:user.Vector{:x 5, :y 5, :z -7}
6
#:user.Vector{:x -267, :y 204, :z -3}Common Lisp
Using the Common Lisp Object System.
<lang lisp>(defclass 3d-vector ()
((x :type number :initarg :x) (y :type number :initarg :y) (z :type number :initarg :z)))
(defmethod print-object ((object 3d-vector) stream)
(print-unreadable-object (object stream :type t)
(with-slots (x y z) object
(format stream "~a ~a ~a" x y z))))
(defun make-3d-vector (x y z)
(make-instance '3d-vector :x x :y y :z z))
(defmethod dot-product ((a 3d-vector) (b 3d-vector))
(with-slots ((a1 x) (a2 y) (a3 z)) a
(with-slots ((b1 x) (b2 y) (b3 z)) b
(+ (* a1 b1) (* a2 b2) (* a3 b3)))))
(defmethod cross-product ((a 3d-vector)
(b 3d-vector))
(with-slots ((a1 x) (a2 y) (a3 z)) a
(with-slots ((b1 x) (b2 y) (b3 z)) b
(make-instance '3d-vector
:x (- (* a2 b3) (* a3 b2))
:y (- (* a3 b1) (* a1 b3))
:z (- (* a1 b2) (* a2 b1))))))
(defmethod scalar-triple-product ((a 3d-vector)
(b 3d-vector)
(c 3d-vector))
(dot-product a (cross-product b c)))
(defmethod vector-triple-product ((a 3d-vector)
(b 3d-vector)
(c 3d-vector))
(cross-product a (cross-product b c)))
(defun vector-products-example ()
(let ((a (make-3d-vector 3 4 5))
(b (make-3d-vector 4 3 5))
(c (make-3d-vector -5 -12 -13)))
(values (dot-product a b)
(cross-product a b)
(scalar-triple-product a b c)
(vector-triple-product a b c))))</lang>
Output:
CL-USER> (vector-products-example) 49 #<3D-VECTOR 5 5 -7> 6 #<3D-VECTOR -267 204 -3>
Using vector type
<lang lisp>(defun cross (a b)
(when (and (equal (length a) 3) (equal (length b) 3))
(vector
(- (* (elt a 1) (elt b 2)) (* (elt a 2) (elt b 1)))
(- (* (elt a 2) (elt b 0)) (* (elt a 0) (elt b 2)))
(- (* (elt a 0) (elt b 1)) (* (elt a 1) (elt b 0))))))
(defun dot (a b)
(when (equal (length a) (length b))
(loop for ai across a for bi across b sum (* ai bi))))
(defun scalar-triple (a b c)
(dot a (cross b c)))
(defun vector-triple (a b c)
(cross a (cross b c)))
(defun task (a b c)
(values (dot a b)
(cross a b)
(scalar-triple a b c)
(vector-triple a b c)))
</lang>
Output:
CL-USER> (task (vector 3 4 5) (vector 4 3 5) (vector -5 -12 -13)) 49 #(5 5 -7) 6 #(-267 204 -3)
Crystal
<lang ruby>class Vector
property x, y, z
def initialize(@x : Int64, @y : Int64, @z : Int64) end
def dot_product(other : Vector)
(self.x * other.x) + (self.y * other.y) + (self.z * other.z)
end
def cross_product(other : Vector)
Vector.new(self.y * other.z - self.z * other.y,
self.z * other.x - self.x * other.z,
self.x * other.y - self.y * other.x)
end
def scalar_triple_product(b : Vector, c : Vector)
self.dot_product(b.cross_product(c))
end
def vector_triple_product(b : Vector, c : Vector)
self.cross_product(b.cross_product(c))
end
def to_s
"(#{self.x}, #{self.y}, #{self.z})\n"
end
end
a = Vector.new(3, 4, 5) b = Vector.new(4, 3, 5) c = Vector.new(-5, -12, -13)
puts "a = #{a.to_s}" puts "b = #{b.to_s}" puts "c = #{c.to_s}" puts "a dot b = #{a.dot_product b}" puts "a cross b = #{a.cross_product(b).to_s}" puts "a dot (b cross c) = #{a.scalar_triple_product b, c}" puts "a cross (b cross c) = #{a.vector_triple_product(b, c).to_s}"</lang>
- Output:
a = (3, 4, 5) b = (4, 3, 5) c = (-5, -12, -13) a dot b = 49 a cross b = (5, 5, -7) a dot (b cross c) = 6 a cross (b cross c) = (-267, 204, -3)
D
<lang d>import std.stdio, std.conv, std.numeric;
struct V3 {
union {
immutable struct { double x, y, z; }
immutable double[3] v;
}
double dot(in V3 rhs) const pure nothrow /*@safe*/ @nogc {
return dotProduct(v, rhs.v);
}
V3 cross(in V3 rhs) const pure nothrow @safe @nogc {
return V3(y * rhs.z - z * rhs.y,
z * rhs.x - x * rhs.z,
x * rhs.y - y * rhs.x);
}
string toString() const { return v.text; }
}
double scalarTriple(in V3 a, in V3 b, in V3 c) /*@safe*/ pure nothrow {
return a.dot(b.cross(c)); // function vector_products.V3.cross (const(V3) rhs) immutable // is not callable using argument types (const(V3)) const
}
V3 vectorTriple(in V3 a, in V3 b, in V3 c) @safe pure nothrow @nogc {
return a.cross(b.cross(c));
}
void main() {
immutable V3 a = {3, 4, 5},
b = {4, 3, 5},
c = {-5, -12, -13};
writeln("a = ", a);
writeln("b = ", b);
writeln("c = ", c);
writeln("a . b = ", a.dot(b));
writeln("a x b = ", a.cross(b));
writeln("a . (b x c) = ", scalarTriple(a, b, c));
writeln("a x (b x c) = ", vectorTriple(a, b, c));
}</lang>
- Output:
a = [3, 4, 5] b = [4, 3, 5] c = [-5, -12, -13] a . b = 49 a x b = [5, 5, -7] a . (b x c) = 6 a x (b x c) = [-267, 204, -3]
EchoLisp
The math library includes the dot-product and cross-product functions. They work on complex or real vectors. <lang scheme> (lib 'math)
(define (scalar-triple-product a b c)
(dot-product a (cross-product b c)))
(define (vector-triple-product a b c)
(cross-product a (cross-product b c)))
(define a #(3 4 5)) (define b #(4 3 5)) (define c #(-5 -12 -13))
(cross-product a b)
→ #( 5 5 -7)
(dot-product a b)
→ 49
(scalar-triple-product a b c)
→ 6
(vector-triple-product a b c)
→ #( -267 204 -3)
</lang>
Elixir
<lang elixir>defmodule Vector do
def dot_product({a1,a2,a3}, {b1,b2,b3}), do: a1*b1 + a2*b2 + a3*b3
def cross_product({a1,a2,a3}, {b1,b2,b3}), do: {a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1}
def scalar_triple_product(a, b, c), do: dot_product(a, cross_product(b, c))
def vector_triple_product(a, b, c), do: cross_product(a, cross_product(b, c))
end
a = {3, 4, 5} b = {4, 3, 5} c = {-5, -12, -13}
IO.puts "a = #{inspect a}" IO.puts "b = #{inspect b}" IO.puts "c = #{inspect c}" IO.puts "a . b = #{inspect Vector.dot_product(a, b)}" IO.puts "a x b = #{inspect Vector.cross_product(a, b)}" IO.puts "a . (b x c) = #{inspect Vector.scalar_triple_product(a, b, c)}" IO.puts "a x (b x c) = #{inspect Vector.vector_triple_product(a, b, c)}"</lang>
- Output:
a = {3, 4, 5}
b = {4, 3, 5}
c = {-5, -12, -13}
a . b = 49
a x b = {5, 5, -7}
a . (b x c) = 6
a x (b x c) = {-267, 204, -3}
Erlang
<lang Erlang> -module(vector). -export([main/0]). vector_product(X,Y)-> [X1,X2,X3]=X, [Y1,Y2,Y3]=Y, Ans=[X2*Y3-X3*Y2,X3*Y1-X1*Y3,X1*Y2-X2*Y1], Ans. dot_product(X,Y)-> [X1,X2,X3]=X, [Y1,Y2,Y3]=Y, Ans=X1*Y1+X2*Y2+X3*Y3, io:fwrite("~p~n",[Ans]). main()-> {ok, A} = io:fread("Enter vector A : ", "~d ~d ~d"), {ok, B} = io:fread("Enter vector B : ", "~d ~d ~d"), {ok, C} = io:fread("Enter vector C : ", "~d ~d ~d"), dot_product(A,B), Ans=vector_product(A,B), io:fwrite("~p,~p,~p~n",Ans), dot_product(C,vector_product(A,B)), io:fwrite("~p,~p,~p~n",vector_product(C,vector_product(A,B))). </lang>
ERRE
<lang ERRE> PROGRAM VECTORPRODUCT
!$DOUBLE
TYPE TVECTOR=(X,Y,Z)
DIM A:TVECTOR,B:TVECTOR,C:TVECTOR
DIM AA:TVECTOR,BB:TVECTOR,CC:TVECTOR DIM DD:TVECTOR,EE:TVECTOR,FF:TVECTOR
PROCEDURE DOTPRODUCT(DD.,EE.->DOTP)
DOTP=DD.X*EE.X+DD.Y*EE.Y+DD.Z*EE.Z
END PROCEDURE
PROCEDURE CROSSPRODUCT(DD.,EE.->FF.)
FF.X=DD.Y*EE.Z-DD.Z*EE.Y FF.Y=DD.Z*EE.X-DD.X*EE.Z FF.Z=DD.X*EE.Y-DD.Y*EE.X
END PROCEDURE
PROCEDURE SCALARTRIPLEPRODUCT(AA.,BB.,CC.->SCALARTP)
CROSSPRODUCT(BB.,CC.->FF.) DOTPRODUCT(AA.,FF.->SCALARTP)
END PROCEDURE
PROCEDURE VECTORTRIPLEPRODUCT(AA.,BB.,CC.->FF.)
CROSSPRODUCT(BB.,CC.->FF.) CROSSPRODUCT(AA.,FF.->FF.)
END PROCEDURE
PROCEDURE PRINTVECTOR(AA.)
PRINT("(";AA.X;",";AA.Y;",";AA.Z;")")
END PROCEDURE
BEGIN
A.X=3 A.Y=4 A.Z=5 B.X=4 B.Y=3 B.Z=5 C.X=-5 C.Y=-12 C.Z=-13
PRINT("A: ";) PRINTVECTOR(A.)
PRINT("B: ";) PRINTVECTOR(B.)
PRINT("C: ";) PRINTVECTOR(C.)
PRINT
DOTPRODUCT(A.,B.->DOTP)
PRINT("A.B =";DOTP)
CROSSPRODUCT(A.,B.->FF.)
PRINT("AxB =";) PRINTVECTOR(FF.)
SCALARTRIPLEPRODUCT(A.,B.,C.->SCALARTP)
PRINT("A.(BxC)=";SCALARTP)
VECTORTRIPLEPRODUCT(A.,B.,C.->FF.)
PRINT("Ax(BxC)=";) PRINTVECTOR(FF.)
END PROGRAM </lang>
Euphoria
<lang euphoria>constant X = 1, Y = 2, Z = 3
function dot_product(sequence a, sequence b)
return a[X]*b[X] + a[Y]*b[Y] + a[Z]*b[Z]
end function
function cross_product(sequence a, sequence b)
return { a[Y]*b[Z] - a[Z]*b[Y],
a[Z]*b[X] - a[X]*b[Z],
a[X]*b[Y] - a[Y]*b[X] }
end function
function scalar_triple(sequence a, sequence b, sequence c)
return dot_product( a, cross_product( b, c ) )
end function
function vector_triple( sequence a, sequence b, sequence c)
return cross_product( a, cross_product( b, c ) )
end function
constant a = { 3, 4, 5 }, b = { 4, 3, 5 }, c = { -5, -12, -13 }
puts(1,"a = ") ? a puts(1,"b = ") ? b puts(1,"c = ") ? c puts(1,"a dot b = ") ? dot_product( a, b ) puts(1,"a x b = ") ? cross_product( a, b ) puts(1,"a dot (b x c) = ") ? scalar_triple( a, b, c ) puts(1,"a x (b x c) = ") ? vector_triple( a, b, c )</lang> Output:
a = {3,4,5}
b = {4,3,5}
c = {-5,-12,-13}
a dot b = 49
a x b = {5,5,-7}
a dot (b x c) = 6
a x (b x c) = {-267,204,-3}
F#
<lang fsharp>let dot (ax, ay, az) (bx, by, bz) =
ax * bx + ay * by + az * bz
let cross (ax, ay, az) (bx, by, bz) =
(ay*bz - az*by, az*bx - ax*bz, ax*by - ay*bx)
let scalTrip a b c =
dot a (cross b c)
let vecTrip a b c =
cross a (cross b c)
[<EntryPoint>] let main _ =
let a = (3.0, 4.0, 5.0) let b = (4.0, 3.0, 5.0) let c = (-5.0, -12.0, -13.0) printfn "%A" (dot a b) printfn "%A" (cross a b) printfn "%A" (scalTrip a b c) printfn "%A" (vecTrip a b c) 0 // return an integer exit code</lang>
- Output:
49.0 (5.0, 5.0, -7.0) 6.0 (-267.0, 204.0, -3.0)
Factor
Factor has a fantastic math.vectors vocabulary, but in the spirit of the task, it is not used. <lang factor>USING: arrays io locals math prettyprint sequences ;
- dot-product ( a b -- dp ) [ * ] 2map sum ;
- cross-product ( a b -- cp )
a first :> a1 a second :> a2 a third :> a3 b first :> b1 b second :> b2 b third :> b3 a2 b3 * a3 b2 * - ! X a3 b1 * a1 b3 * - ! Y a1 b2 * a2 b1 * - ! Z 3array ;
- scalar-triple-product ( a b c -- stp )
cross-product dot-product ;
- vector-triple-product ( a b c -- vtp )
cross-product cross-product ;
[let
{ 3 4 5 } :> a
{ 4 3 5 } :> b
{ -5 -12 -13 } :> c
"a: " write a .
"b: " write b .
"c: " write c . nl
"a . b: " write a b dot-product .
"a x b: " write a b cross-product .
"a . (b x c): " write a b c scalar-triple-product .
"a x (b x c): " write a b c vector-triple-product .
]</lang>
- Output:
a: { 3 4 5 }
b: { 4 3 5 }
c: { -5 -12 -13 }
a . b: 49
a x b: { 5 5 -7 }
a . (b x c): 6
a x (b x c): { -267 204 -3 }
Fantom
<lang fantom>class Main {
Int dot_product (Int[] a, Int[] b)
{
a[0]*b[0] + a[1]*b[1] + a[2]*b[2]
}
Int[] cross_product (Int[] a, Int[] b)
{
[a[1]*b[2] - a[2]*b[1], a[2]*b[0] - a[0]*b[2], a[0]*b[1]-a[1]*b[0]]
}
Int scalar_triple_product (Int[] a, Int[] b, Int[] c)
{
dot_product (a, cross_product (b, c))
}
Int[] vector_triple_product (Int[] a, Int[] b, Int[] c)
{
cross_product (a, cross_product (b, c))
}
Void main ()
{
a := [3, 4, 5]
b := [4, 3, 5]
c := [-5, -12, -13]
echo ("a . b = " + dot_product (a, b))
echo ("a x b = [" + cross_product(a, b).join (", ") + "]")
echo ("a . (b x c) = " + scalar_triple_product (a, b, c))
echo ("a x (b x c) = [" + vector_triple_product(a, b, c).join (", ") + "]")
}
}</lang> Output:
a . b = 49 a x b = [5, 5, -7] a . (b x c) = 6 a x (b x c) = [-267, 204, -3]
Forth
<lang Forth>
- 3f! ( &v - ) ( f: x y z - ) dup float+ dup float+ f! f! f! ;
- Vector \ Compiletime: ( f: x y z - ) ( <name> - )
create here [ 3 floats ] literal allot 3f! ; \ Runtime: ( - &v )
- >fx@ ( &v - ) ( f: - n ) postpone f@ ; immediate
- >fy@ ( &v - ) ( f: - n ) float+ f@ ;
- >fz@ ( &v - ) ( f: - n ) float+ float+ f@ ;
- .Vector ( &v - ) dup >fz@ dup >fy@ >fx@ f. f. f. ;
- Dot* ( &v1 &v2 - ) ( f - DotPrd )
2dup >fx@ >fx@ f*
2dup >fy@ >fy@ f* f+
>fz@ >fz@ f* f+ ;
- Cross* ( &v1 &v2 &vResult - )
>r 2dup >fz@ >fy@ f*
2dup >fy@ >fz@ f* f-
2dup >fx@ >fz@ f*
2dup >fz@ >fx@ f* f-
2dup >fy@ >fx@ f*
>fx@ >fy@ f* f-
r> 3f! ;
- ScalarTriple* ( &v1 &v2 &v3 - ) ( f: - ScalarTriple* )
>r pad Cross* pad r> Dot* ;
- VectorTriple* ( &v1 &v2 &v3 &vDest - )
>r swap r@ Cross* r> tuck Cross* ;
3e 4e 5e Vector A 4e 3e 5e Vector B
-5e -12e -13e Vector C
cr cr .( a . b = ) A B Dot* f. cr .( a x b = ) A B pad Cross* pad .Vector cr .( a . [b x c] = ) A B C ScalarTriple* f. cr .( a x [b x c] = ) A B C pad VectorTriple* pad .Vector </lang>
- Output:
a . b = 49.0000 a x b = 5.00000 5.00000 -7.00000 a . [b x c] = 6.00000 a x [b x c] = -267.000 204.000 -3.00000
<lang forth>
S" fsl-util.fs" REQUIRED
- 3f! 3 SWAP }fput ;
- vector
CREATE HERE 3 DUP FLOAT DUP , * ALLOT SWAP CELL+ }fput DOES> CELL+ ;
- >fx@ 0 } F@ ;
- >fy@ 1 } F@ ;
- >fz@ 2 } F@ ;
- .Vector 3 SWAP }fprint ;
0e 0e 0e vector pad \ NB: your system will be non-standard after this line
\ From here on is identical to the above example</lang>
Fortran
Specialized for 3-dimensional vectors. <lang fortran>program VectorProducts
real, dimension(3) :: a, b, c
a = (/ 3, 4, 5 /) b = (/ 4, 3, 5 /) c = (/ -5, -12, -13 /)
print *, dot_product(a, b) print *, cross_product(a, b) print *, s3_product(a, b, c) print *, v3_product(a, b, c)
contains
function cross_product(a, b) real, dimension(3) :: cross_product real, dimension(3), intent(in) :: a, b
cross_product(1) = a(2)*b(3) - a(3)*b(2) cross_product(2) = a(3)*b(1) - a(1)*b(3) cross_product(3) = a(1)*b(2) - b(1)*a(2) end function cross_product
function s3_product(a, b, c) real :: s3_product real, dimension(3), intent(in) :: a, b, c
s3_product = dot_product(a, cross_product(b, c)) end function s3_product
function v3_product(a, b, c) real, dimension(3) :: v3_product real, dimension(3), intent(in) :: a, b, c
v3_product = cross_product(a, cross_product(b, c)) end function v3_product
end program VectorProducts</lang> Output
49.0000
5.00000 5.00000 -7.00000
6.00000
-267.000 204.000 -3.00000
FreeBASIC
<lang FreeBASIC> 'Construct only required operators for this. Type V3
As double x,y,z declare operator cast() as string
End Type
- define dot *
- define cross ^
- define Show(t1,t) ? #t1;tab(22);t
operator V3.cast() as string return "("+str(x)+","+str(y)+","+str(z)+")" end operator
Operator dot(v1 As v3,v2 As v3) As double Return v1.x*v2.x+v1.y*v2.y+v1.z*v2.z End Operator
Operator cross(v1 As v3,v2 As v3) As v3 Return type<v3>(v1.y*v2.z-v2.y*v1.z,-(v1.x*v2.z-v2.x*v1.z),v1.x*v2.y-v2.x*v1.y) End Operator
dim as V3 a = (3, 4, 5), b = (4, 3, 5), c = (-5, -12, -13)
Show(a,a) Show(b,b) Show(c,c) ? Show(a . b,a dot b) Show(a X b,a cross b) Show(a . b X c,a dot b cross c) Show(a X (b X c),a cross (b cross c)) sleep</lang>
- Output:
a (3,4,5) b (4,3,5) c (-5,-12,-13) a . b 49 a X b (5,5,-7) a . b X c 6 a X (b X c) (-267,204,-3)
FunL
<lang funl>A = (3, 4, 5) B = (4, 3, 5) C = (-5, -12, -13)
def dot( u, v ) = sum( u(i)v(i) | i <- 0:u.>length() ) def cross( u, v ) = (u(1)v(2) - u(2)v(1), u(2)v(0) - u(0)v(2), u(0)v(1) - u(1)v(0) ) def scalarTriple( u, v, w ) = dot( u, cross(v, w) ) def vectorTriple( u, v, w ) = cross( u, cross(v, w) )
println( "A\u00b7B = ${dot(A, B)}" ) println( "A\u00d7B = ${cross(A, B)}" ) println( "A\u00b7(B\u00d7C) = ${scalarTriple(A, B, C)}" ) println( "A\u00d7(B\u00d7C) = ${vectorTriple(A, B, C)}" )</lang>
- Output:
A·B = 49 A×B = (5, 5, -7) A·(B×C) = 6 A×(B×C) = (-267, 204, -3)
GAP
<lang gap>DotProduct := function(u, v) return u*v; end;
CrossProduct := function(u, v) return [ u[2]*v[3] - u[3]*v[2], u[3]*v[1] - u[1]*v[3], u[1]*v[2] - u[2]*v[1] ]; end;
ScalarTripleProduct := function(u, v, w) return DotProduct(u, CrossProduct(v, w)); end;
VectorTripleProduct := function(u, v, w) return CrossProduct(u, CrossProduct(v, w)); end;
a := [3, 4, 5]; b := [4, 3, 5]; c := [-5, -12, -13];
DotProduct(a, b);
- 49
CrossProduct(a, b);
- [ 5, 5, -7 ]
ScalarTripleProduct(a, b, c);
- 6
- Another way to get it
Determinant([a, b, c]);
- 6
VectorTripleProduct(a, b, c);
- [ -267, 204, -3 ]</lang>
GLSL
<lang glsl> vec3 a = vec3(3, 4, 5),b = vec3(4, 3, 5),c = vec3(-5, -12, -13);
float dotProduct(vec3 a, vec3 b) { return a.x*b.x+a.y*b.y+a.z*b.z; }
vec3 crossProduct(vec3 a,vec3 b) { vec3 c = vec3(a.y*b.z - a.z*b.y, a.z*b.x - a.x*b.z, a.x*b.y- a.y*b.x);
return c; }
float scalarTripleProduct(vec3 a,vec3 b,vec3 c) { return dotProduct(a,crossProduct(b,c)); }
vec3 vectorTripleProduct(vec3 a,vec3 b,vec3 c) { return crossProduct(a,crossProduct(b,c)); } </lang>
Go
<lang go>package main
import "fmt"
type vector struct {
x, y, z float64
}
var (
a = vector{3, 4, 5}
b = vector{4, 3, 5}
c = vector{-5, -12, -13}
)
func dot(a, b vector) float64 {
return a.x*b.x + a.y*b.y + a.z*b.z
}
func cross(a, b vector) vector {
return vector{a.y*b.z - a.z*b.y, a.z*b.x - a.x*b.z, a.x*b.y - a.y*b.x}
}
func s3(a, b, c vector) float64 {
return dot(a, cross(b, c))
}
func v3(a, b, c vector) vector {
return cross(a, cross(b, c))
}
func main() {
fmt.Println(dot(a, b)) fmt.Println(cross(a, b)) fmt.Println(s3(a, b, c)) fmt.Println(v3(a, b, c))
}</lang> Output:
49
{5 5 -7}
6
{-267 204 -3}
Groovy
Dot Product Solution: <lang groovy>def pairwiseOperation = { x, y, Closure binaryOp ->
assert x && y && x.size() == y.size() [x, y].transpose().collect(binaryOp)
}
def pwMult = pairwiseOperation.rcurry { it[0] * it[1] }
def dotProduct = { x, y ->
assert x && y && x.size() == y.size() pwMult(x, y).sum()
}</lang> Cross Product Solution, using scalar operations: <lang groovy>def crossProductS = { x, y ->
assert x && y && x.size() == 3 && y.size() == 3 [x[1]*y[2] - x[2]*y[1], x[2]*y[0] - x[0]*y[2] , x[0]*y[1] - x[1]*y[0]]
}</lang> Cross Product Solution, using "vector" operations: <lang groovy>def rotR = {
assert it && it.size() > 2 [it[-1]] + it[0..-2]
}
def rotL = {
assert it && it.size() > 2 it[1..-1] + [it[0]]
}
def pwSubtr = pairwiseOperation.rcurry { it[0] - it[1] }
def crossProductV = { x, y ->
assert x && y && x.size() == 3 && y.size() == 3 pwSubtr(pwMult(rotL(x), rotR(y)), pwMult(rotL(y), rotR(x)))
}</lang> Test program (including triple products): <lang groovy>def test = { crossProduct ->
def scalarTripleProduct = { x, y, z ->
dotProduct(x, crossProduct(y, z))
}
def vectorTripleProduct = { x, y, z ->
crossProduct(x, crossProduct(y, z))
}
def a = [3, 4, 5] def b = [4, 3, 5] def c = [-5, -12, -13]
println(" a . b = " + dotProduct(a,b))
println(" a x b = " + crossProduct(a,b))
println("a . (b x c) = " + scalarTripleProduct(a,b,c))
println("a x (b x c) = " + vectorTripleProduct(a,b,c))
println()
}
test(crossProductS) test(crossProductV)</lang> Output:
a . b = 49
a x b = [5, 5, -7]
a . (b x c) = 6
a x (b x c) = [-267, 204, -3]
a . b = 49
a x b = [5, 5, -7]
a . (b x c) = 6
a x (b x c) = [-267, 204, -3]Haskell
<lang haskell>import Data.Monoid ((<>))
type Vector a = [a]
type Scalar a = a
a, b, c, d :: Vector Int a = [3, 4, 5]
b = [4, 3, 5]
c = [-5, -12, -13]
d = [3, 4, 5, 6]
dot
:: (Num t) => Vector t -> Vector t -> Scalar t
dot u v
| length u == length v = sum $ zipWith (*) u v | otherwise = error "Dotted Vectors must be of equal dimension."
cross
:: (Num t) => Vector t -> Vector t -> Vector t
cross u v
| length u == 3 && length v == 3 = [ u !! 1 * v !! 2 - u !! 2 * v !! 1 , u !! 2 * head v - head u * v !! 2 , head u * v !! 1 - u !! 1 * head v ] | otherwise = error "Crossed Vectors must both be three dimensional."
scalarTriple
:: (Num t) => Vector t -> Vector t -> Vector t -> Scalar t
scalarTriple q r s = dot q $ cross r s
vectorTriple
:: (Num t) => Vector t -> Vector t -> Vector t -> Vector t
vectorTriple q r s = cross q $ cross r s
main :: IO () main =
mapM_ putStrLn [ "a . b = " <> show (dot a b) , "a x b = " <> show (cross a b) , "a . b x c = " <> show (scalarTriple a b c) , "a x b x c = " <> show (vectorTriple a b c) , "a . d = " <> show (dot a d) ]</lang>
Output:
a . b = 49 a x b = [5,5,-7] a . b x c = 6 a x b x c = [-267,204,-3] ** Exception: Dotted Vectors must be of equal dimension. a . d =
Or using Either and (>>=), rather than error, to pass on intelligible messages:
<lang haskell>dotProduct
:: Num a => [a] -> [a] -> Either String a
dotProduct xs ys
| length xs /= length ys = Left "Dot product not defined - vectors differ in dimension." | otherwise = Right (sum $ zipWith (*) xs ys)
crossProduct
:: Num a => [a] -> [a] -> Either String [a]
crossProduct xs ys
| 3 /= length xs || 3 /= length ys = Left "crossProduct is defined only for 3d vectors." | otherwise = Right [x2 * y3 - x3 * y2, x3 * y1 - x1 * y3, x1 * y2 - x2 * y1] where [x1, x2, x3] = xs [y1, y2, y3] = ys
scalarTriple
:: Num a => [a] -> [a] -> [a] -> Either String a
scalarTriple q r s = crossProduct r s >>= dotProduct q
vectorTriple
:: Num a => [a] -> [a] -> [a] -> Either String [a]
vectorTriple q r s = crossProduct r s >>= crossProduct q
-- TEST --------------------------------------------------- a = [3, 4, 5]
b = [4, 3, 5]
c = [-5, -12, -13]
d = [3, 4, 5, 6]
main :: IO () main =
mapM_ putStrLn $ zipWith (++) ["a . b", "a x b", "a . b x c", "a x b x c", "a . d", "a . (b x d)"] [ sh $ dotProduct a b , sh $ crossProduct a b , sh $ scalarTriple a b c , sh $ vectorTriple a b c , sh $ dotProduct a d , sh $ scalarTriple a b d ]
sh
:: Show a => Either String a -> String
sh = either (" => " ++) ((" = " ++) . show)</lang>
- Output:
a . b = 49 a x b = [5,5,-7] a . b x c = 6 a x b x c = [-267,204,-3] a . d => Dot product not defined - vectors differ in dimension. a . (b x d) => crossProduct is defined only for 3d vectors.
Icon and Unicon
<lang icon># record type to store a 3D vector record Vector3D(x, y, z)
- procedure to display vector as a string
procedure toString (vector)
return "(" || vector.x || ", " || vector.y || ", " || vector.z || ")"
end
procedure dotProduct (a, b)
return a.x * b.x + a.y * b.y + a.z * b.z
end
procedure crossProduct (a, b)
x := a.y * b.z - a.z * b.y y := a.z * b.x - a.x * b.z z := a.x * b.y - a.y * b.x return Vector3D(x, y, z)
end
procedure scalarTriple (a, b, c)
return dotProduct (a, crossProduct (b, c))
end
procedure vectorTriple (a, b, c)
return crossProduct (a, crossProduct (b, c))
end
- main procedure, to run given test
procedure main ()
a := Vector3D(3, 4, 5) b := Vector3D(4, 3, 5) c := Vector3D(-5, -12, -13)
writes ("A.B : " || toString(a) || "." || toString(b) || " = ")
write (dotProduct (a, b))
writes ("AxB : " || toString(a) || "x" || toString(b) || " = ")
write (toString(crossProduct (a, b)))
writes ("A.(BxC) : " || toString(a) || ".(" || toString(b) || "x" || toString(c) || ") = ")
write (scalarTriple (a, b, c))
writes ("Ax(BxC) : " || toString(a) || "x(" || toString(b) || "x" || toString(c) || ") = ")
write (toString(vectorTriple (a, b, c)))
end</lang> Output:
A.B : (3, 4, 5).(4, 3, 5) = 49 AxB : (3, 4, 5)x(4, 3, 5) = (5, 5, -7) A.(BxC) : (3, 4, 5).((4, 3, 5)x(-5, -12, -13)) = 6 Ax(BxC) : (3, 4, 5)x((4, 3, 5)x(-5, -12, -13)) = (-267, 204, -3)
J
Perhaps the most straightforward definition for cross product in J uses rotate multiply and subtract:
<lang j>cross=: (1&|.@[ * 2&|.@]) - 2&|.@[ * 1&|.@]</lang>
However, there are other valid approaches. For example, a "generalized approach" based on j:Essays/Complete Tensor: <lang j>CT=: C.!.2 @ (#:i.) @ $~ ip=: +/ .* NB. inner product cross=: ] ip CT@#@[ ip [</lang>
Note that there are a variety of other generalizations have cross products as a part of what they do.
An alternative definition for cross (based on finding the determinant of a 3 by 3 matrix where one row is unit vectors) could be: <lang j>cross=: [: > [: -&.>/ .(*&.>) (<"1=i.3) , ,:&:(<"0)</lang>
With an implementation of cross product and inner product, the rest of the task becomes trivial:
<lang j>a=: 3 4 5 b=: 4 3 5 c=: -5 12 13
A=: 0 {:: ] NB. contents of the first box on the right B=: 1 {:: ] NB. contents of the second box on the right C=: 2 {:: ] NB. contents of the third box on the right
dotP=: A ip B crossP=: A cross B scTriP=: A ip B cross C veTriP=: A cross B cross C</lang> Required example: <lang j> dotP a;b 49
crossP a;b
5 5 _7
scTriP a;b;c
6
veTriP a;b;c
_267 204 _3</lang>
Java
All operations which return vectors give vectors containing Doubles.
<lang java5>public class VectorProds{
public static class Vector3D<T extends Number>{
private T a, b, c;
public Vector3D(T a, T b, T c){
this.a = a;
this.b = b;
this.c = c;
}
public double dot(Vector3D<?> vec){
return (a.doubleValue() * vec.a.doubleValue() +
b.doubleValue() * vec.b.doubleValue() +
c.doubleValue() * vec.c.doubleValue());
}
public Vector3D<Double> cross(Vector3D<?> vec){
Double newA = b.doubleValue()*vec.c.doubleValue() - c.doubleValue()*vec.b.doubleValue();
Double newB = c.doubleValue()*vec.a.doubleValue() - a.doubleValue()*vec.c.doubleValue();
Double newC = a.doubleValue()*vec.b.doubleValue() - b.doubleValue()*vec.a.doubleValue();
return new Vector3D<Double>(newA, newB, newC);
}
public double scalTrip(Vector3D<?> vecB, Vector3D<?> vecC){
return this.dot(vecB.cross(vecC));
}
public Vector3D<Double> vecTrip(Vector3D<?> vecB, Vector3D<?> vecC){
return this.cross(vecB.cross(vecC));
}
@Override
public String toString(){
return "<" + a.toString() + ", " + b.toString() + ", " + c.toString() + ">";
}
}
public static void main(String[] args){
Vector3D<Integer> a = new Vector3D<Integer>(3, 4, 5);
Vector3D<Integer> b = new Vector3D<Integer>(4, 3, 5);
Vector3D<Integer> c = new Vector3D<Integer>(-5, -12, -13);
System.out.println(a.dot(b));
System.out.println(a.cross(b));
System.out.println(a.scalTrip(b, c));
System.out.println(a.vecTrip(b, c));
}
}</lang> Output:
49.0 <5.0, 5.0, -7.0> 6.0 <-267.0, 204.0, -3.0>
This solution uses Java SE new Stream API <lang Java8>import java.util.Arrays; import java.util.stream.IntStream;
public class VectorsOp { // Vector dot product using Java SE 8 stream abilities // the method first create an array of size values, // and map the product of each vectors components in a new array (method map()) // and transform the array to a scalr by summing all elements (method reduce) // the method parallel is there for optimization private static int dotProduct(int[] v1, int[] v2,int length) {
int result = IntStream.range(0, length) .parallel() .map( id -> v1[id] * v2[id]) .reduce(0, Integer::sum);
return result; }
// Vector Cross product using Java SE 8 stream abilities // here we map in a new array where each element is equal to the cross product // With Stream is is easier to handle N dimensions vectors private static int[] crossProduct(int[] v1, int[] v2,int length) {
int result[] = new int[length] ; //result[0] = v1[1] * v2[2] - v1[2]*v2[1] ; //result[1] = v1[2] * v2[0] - v1[0]*v2[2] ; // result[2] = v1[0] * v2[1] - v1[1]*v2[0] ;
result = IntStream.range(0, length) .parallel() .map( i -> v1[(i+1)%length] * v2[(i+2)%length] - v1[(i+2)%length]*v2[(i+1)%length]) .toArray();
return result; }
public static void main (String[] args) { int[] vect1 = {3, 4, 5}; int[] vect2 = {4, 3, 5}; int[] vect3 = {-5, -12, -13};
System.out.println("dot product =:" + dotProduct(vect1,vect2,3));
int[] prodvect = new int[3]; prodvect = crossProduct(vect1,vect2,3); System.out.println("cross product =:[" + prodvect[0] + "," + prodvect[1] + "," + prodvect[2] + "]");
prodvect = crossProduct(vect2,vect3,3); System.out.println("scalar product =:" + dotProduct(vect1,prodvect,3));
prodvect = crossProduct(vect1,prodvect,3);
System.out.println("triple product =:[" + prodvect[0] + "," + prodvect[1] + "," + prodvect[2] + "]");
} }</lang> result is the same as above , fortunately
dot product =:49 cross product =:[5,5,-7] scalar product =:6 triple product =:[-267,204,-3]
JavaScript
ES5
The dotProduct() function is generic and will create a dot product of any set of vectors provided they are all the same dimension.
The crossProduct() function expects two 3D vectors.
<lang javascript>function dotProduct() {
var len = arguments[0] && arguments[0].length;
var argsLen = arguments.length;
var i, j = len;
var prod, sum = 0;
// If no arguments supplied, return undefined
if (!len) {
return;
}
// If all vectors not same length, return undefined
i = argsLen;
while (i--) {
if (arguments[i].length != len) {
return; // return undefined
}
}
// Sum terms
while (j--) {
i = argsLen;
prod = 1;
while (i--) {
prod *= arguments[i][j];
}
sum += prod;
}
return sum;
}
function crossProduct(a, b) {
// Check lengths
if (a.length != 3 || b.length != 3) {
return;
}
return [a[1]*b[2] - a[2]*b[1],
a[2]*b[0] - a[0]*b[2],
a[0]*b[1] - a[1]*b[0]];
}
function scalarTripleProduct(a, b, c) {
return dotProduct(a, crossProduct(b, c));
}
function vectorTripleProduct(a, b, c) {
return crossProduct(a, crossProduct(b, c));
}
// Run tests (function () {
var a = [3, 4, 5]; var b = [4, 3, 5]; var c = [-5, -12, -13]; alert( 'A . B: ' + dotProduct(a, b) + '\n' + 'A x B: ' + crossProduct(a, b) + '\n' + 'A . (B x C): ' + scalarTripleProduct(a, b, c) + '\n' + 'A x (B x C): ' + vectorTripleProduct(a, b, c) );
}());</lang>
- Output:
A . B: 49 A x B: 5,5,-7 A . (B x C): 6 A x (B x C): -267,204,-3
ES6
<lang javascript>(() => {
'use strict';
// dotProduct :: [a] -> [a] -> Either String a
const dotProduct = xs =>
// Dot product of two vectors of equal dimension.
ys => xs.length !== ys.length ? (
Left('Dot product not defined - vectors differ in dimension.')
) : Right(sum(
zipWith(mul)(Array.from(xs))(Array.from(ys))
));
// crossProduct :: Num a => (a, a, a) -> (a, a, a)
// Either String -> (a, a, a)
const crossProduct = xs =>
// Cross product of two 3D vectors.
ys => 3 !== xs.length || 3 !== ys.length ? (
Left('crossProduct is defined only for 3d vectors.')
) : Right((() => {
const [x1, x2, x3] = Array.from(xs);
const [y1, y2, y3] = Array.from(ys);
return [
x2 * y3 - x3 * y2,
x3 * y1 - x1 * y3,
x1 * y2 - x2 * y1
];
})());
// scalarTriple :: Num a => (a, a, a) -> (a, a, a) -> (a, a a) ->
// Either String -> a
const scalarTriple = q =>
// The scalar triple product.
r => s => bindLR(crossProduct(r)(s))(
dotProduct(q)
);
// vectorTriple :: Num a => (a, a, a) -> (a, a, a) -> (a, a a) ->
// Either String -> (a, a, a)
const vectorTriple = q =>
// The vector triple product.
r => s => bindLR(crossProduct(r)(s))(
crossProduct(q)
);
// main :: IO ()
const main = () => {
// TEST -------------------------------------------
const
a = [3, 4, 5],
b = [4, 3, 5],
c = [-5, -12, -13],
d = [3, 4, 5, 6];
console.log(unlines(
zipWith(k => f => k + show(
saturated(f)([a, b, c])
))(['a . b', 'a x b', 'a . (b x c)', 'a x (b x c)'])(
[dotProduct, crossProduct, scalarTriple, vectorTriple]
)
.concat([
'a . d' + show(
dotProduct(a)(d)
),
'a . (b x d)' + show(
scalarTriple(a)(b)(d)
)
])
));
};
// GENERIC FUNCTIONS ----------------------------------
// Left :: a -> Either a b
const Left = x => ({
type: 'Either',
Left: x
});
// Right :: b -> Either a b
const Right = x => ({
type: 'Either',
Right: x
});
// bindLR (>>=) :: Either a -> (a -> Either b) -> Either b
const bindLR = m => mf =>
undefined !== m.Left ? (
m
) : mf(m.Right);
// either :: (a -> c) -> (b -> c) -> Either a b -> c
const either = fl => fr => e =>
'Either' === e.type ? (
undefined !== e.Left ? (
fl(e.Left)
) : fr(e.Right)
) : undefined;
// identity :: a -> a const identity = x => x;
// mul (*) :: Num a => a -> a -> a const mul = a => b => a * b;
// Curried function -> [Argument] -> a more saturated value
const saturated = f =>
// A curried function applied successively to
// a list of arguments up to, but not beyond,
// the point of saturation.
args => 0 < args.length ? (
args.slice(1).reduce(
(a, x) => 'function' !== typeof a ? (
a
) : a(x),
f(args[0])
)
) : f;
// show :: Either String a -> String
const show = x =>
either(x => ' => ' + x)(
x => ' = ' + JSON.stringify(x)
)(x);
// sum :: [Num] -> Num const sum = xs => xs.reduce((a, x) => a + x, 0);
// unlines :: [String] -> String
const unlines = xs => xs.join('\n');
// zipWith:: (a -> b -> c) -> [a] -> [b] -> [c]
const zipWith = f => xs => ys =>
xs.slice(
0, Math.min(xs.length, ys.length)
).map((x, i) => f(x)(ys[i]));
// MAIN --- return main();
})();</lang>
- Output:
a . b = 49 a x b = [5,5,-7] a . (b x c) = 6 a x (b x c) = [-267,204,-3] a . d => Dot product not defined - vectors differ in dimension. a . (b x d) => crossProduct is defined only for 3d vectors.
jq
The dot_product() function is generic and will create a dot product of any pair of vectors provided they are both the same dimension. The other functions expect 3D vectors.<lang jq>def dot_product(a; b):
reduce range(0;a|length) as $i (0; . + (a[$i] * b[$i]) );
- for 3d vectors
def cross_product(a;b):
[ a[1]*b[2] - a[2]*b[1], a[2]*b[0] - a[0]*b[2], a[0]*b[1]-a[1]*b[0] ];
def scalar_triple_product(a;b;c):
dot_product(a; cross_product(b; c));
def vector_triple_product(a;b;c):
cross_product(a; cross_product(b; c));
def main:
[3, 4, 5] as $a
| [4, 3, 5] as $b
| [-5, -12, -13] as $c
| "a . b = \(dot_product($a; $b))",
"a x b = [\( cross_product($a; $b) | map(tostring) | join (", ") )]" ,
"a . (b x c) = \( scalar_triple_product ($a; $b; $c)) )",
"a x (b x c) = [\( vector_triple_product($a; $b; $c)|map(tostring)|join (", ") )]" ;</lang>
Output: <lang jq>"a . b = 49" "a x b = [5, 5, -7]" "a . (b x c) = 6 )" "a x (b x c) = [-267, 204, -3]"</lang>
Julia
Julia provides dot and cross products with LinearAlgebra. It's easy enough to use these to construct the triple products. <lang julia>using LinearAlgebra
function scalarproduct(a::AbstractVector{T}, b::AbstractVector{T}, c::AbstractVector{T}) where {T<:Number}
return dot(a, cross(b, c))
end
function vectorproduct(a::AbstractVector{T}, b::AbstractVector{T}, c::AbstractVector{T}) where {T<:Number}
return cross(a, cross(b, c))
end
const a = [3, 4, 5] const b = [4, 3, 5] const c = [-5, -12, -13]
println("Test Vectors:") @show a b c
println("\nVector Products:") @show dot(a, b) @show cross(a, b) @show scalarproduct(a, b, c) @show vectorproduct(a, b, c)</lang>
- Output:
Test Vectors: a = [3, 4, 5] b = [4, 3, 5] c = [-5, -12, -13] Vector Products: dot(a, b) = 49 cross(a, b) = [5, 5, -7] scalarproduct(a, b, c) = 6 vectorproduct(a, b, c) = [-267, 204, -3]
Kotlin
<lang scala>// version 1.1.2
class Vector3D(val x: Double, val y: Double, val z: Double) {
infix fun dot(v: Vector3D) = x * v.x + y * v.y + z * v.z
infix fun cross(v: Vector3D) =
Vector3D(y * v.z - z * v.y, z * v.x - x * v.z, x * v.y - y * v.x)
fun scalarTriple(v: Vector3D, w: Vector3D) = this dot (v cross w)
fun vectorTriple(v: Vector3D, w: Vector3D) = this cross (v cross w)
override fun toString() = "($x, $y, $z)"
}
fun main(args: Array<String>) {
val a = Vector3D(3.0, 4.0, 5.0)
val b = Vector3D(4.0, 3.0, 5.0)
val c = Vector3D(-5.0, -12.0, -13.0)
println("a = $a")
println("b = $b")
println("c = $c")
println()
println("a . b = ${a dot b}")
println("a x b = ${a cross b}")
println("a . b x c = ${a.scalarTriple(b, c)}")
println("a x b x c = ${a.vectorTriple(b, c)}")
}</lang>
- Output:
a = (3.0, 4.0, 5.0) b = (4.0, 3.0, 5.0) c = (-5.0, -12.0, -13.0) a . b = 49.0 a x b = (5.0, 5.0, -7.0) a . b x c = 6.0 a x b x c = (-267.0, 204.0, -3.0)
Liberty BASIC
<lang lb> print "Vector products of 3-D vectors"
print "Dot product of 3,4,5 and 4,3,5 is " print DotProduct( "3,4,5", "4,3,5") print "Cross product of 3,4,5 and 4,3,5 is " print CrossProduct$( "3,4,5", "4,3,5") print "Scalar triple product of 3,4,5, 4,3,5 -5, -12, -13 is " print ScalarTripleProduct( "3,4,5", "4,3,5", "-5, -12, -13") print "Vector triple product of 3,4,5, 4,3,5 -5, -12, -13 is " print VectorTripleProduct$( "3,4,5", "4,3,5", "-5, -12, -13")
end
function DotProduct( i$, j$)
ix =val( word$( i$, 1, ","))
iy =val( word$( i$, 2, ","))
iz =val( word$( i$, 3, ","))
jx =val( word$( j$, 1, ","))
jy =val( word$( j$, 2, ","))
jz =val( word$( j$, 3, ","))
DotProduct = ix *jx +iy *jy + iz *jz
end function
function CrossProduct$( i$, j$)
ix =val( word$( i$, 1, ","))
iy =val( word$( i$, 2, ","))
iz =val( word$( i$, 3, ","))
jx =val( word$( j$, 1, ","))
jy =val( word$( j$, 2, ","))
jz =val( word$( j$, 3, ","))
cpx =iy *jz -iz *jy
cpy =iz *jx -ix *jz
cpz =ix *jy -iy *jx
CrossProduct$ =str$( cpx); ","; str$( cpy); ","; str$( cpz)
end function
function ScalarTripleProduct( i$, j$, k$))
ScalarTripleProduct =DotProduct( i$, CrossProduct$( j$, k$))
end function
function VectorTripleProduct$( i$, j$, k$))
VectorTripleProduct$ =CrossProduct$( i$, CrossProduct$( j$, k$))
end function
END SUB</lang>
Lingo
Lingo has a built-in vector data type that supports calculation of both dot and cross products: <lang lingo>a = vector(1,2,3) b = vector(4,5,6)
put a * b -- 32.0000
put a.dot(b) -- 32.0000
put a.cross(b) -- vector( -3.0000, 6.0000, -3.0000 )</lang>
Lua
<lang lua>Vector = {} function Vector.new( _x, _y, _z )
return { x=_x, y=_y, z=_z }
end
function Vector.dot( A, B )
return A.x*B.x + A.y*B.y + A.z*B.z
end
function Vector.cross( A, B )
return { x = A.y*B.z - A.z*B.y,
y = A.z*B.x - A.x*B.z,
z = A.x*B.y - A.y*B.x }
end
function Vector.scalar_triple( A, B, C )
return Vector.dot( A, Vector.cross( B, C ) )
end
function Vector.vector_triple( A, B, C )
return Vector.cross( A, Vector.cross( B, C ) )
end
A = Vector.new( 3, 4, 5 )
B = Vector.new( 4, 3, 5 )
C = Vector.new( -5, -12, -13 )
print( Vector.dot( A, B ) )
r = Vector.cross(A, B ) print( r.x, r.y, r.z )
print( Vector.scalar_triple( A, B, C ) )
r = Vector.vector_triple( A, B, C ) print( r.x, r.y, r.z )</lang>
49 5 5 -7 6 -267 204 -3
M2000 Interpreter
<lang M2000 Interpreter> Module checkit {
class Vector {
\\ by default are double
a,b,c
Property ToString$ {
Value {
link parent a,b,c to a,b,c
value$=format$("({0}, {1}, {2})",a,b,c)
}
}
Operator "==" {
read n
push .a==n.a and .b==n.b and .c==n.c
}
Operator Unary {
.a-! : .b-! : .c-!
}
Operator "+" {
Read v2
For this, v2 {
.a+=..a :.b+=..b:.c+=..c:
}
}
Function Mul(r) {
vv=this
for vv {
.a*=r:.b*=r:.c*=r
}
=vv
}
Function Dot(v2) {
def double sum
for this, v2 {
sum=.a*..a+.b*..b+.c*..c
}
=sum
}
Operator "*" {
Read v2
For This, v2 {
Push .b*..c-.c*..b
Push .c*..a-.a*..c
.c<=.a*..b-.b*..a
Read .b, .a
}
}
class:
module Vector {
if match("NNN") then {
Read .a,.b,.c
}
}
}
A=Vector(3,4,5)
B=Vector(4,3,5)
C=Vector(-5,-12,-13)
Print "A=";A.toString$
Print "B=";B.toString$
Print "C=";C.toString$
Print "A dot B="; A.dot(B)
AxB=A*B
Print "A x B="; AxB.toString$
Print "A dot (B x C)=";A.dot(B*C)
AxBxC=A*(B*C)
Print "A x (B x C)=";AxBxC.toString$
Def ToString$(a)=a.toString$
Print "A x (B x C)=";ToString$(A*(B*C))
} Checkit </lang>
- Output:
A=(3, 4, 5) B=(4, 3, 5) C=(-5, -12, -13) A dot B=49 A x B=(5, 5, -7) A dot (B x C)=6 A x (B x C)=(-267, 204, -3) A x (B x C)=(-267, 204, -3)
Maple
<lang Maple>with(LinearAlgebra): A := Vector([3,4,5]): B := Vector([4,3,5]): C := Vector([-5,-12,-13]): >>>A.B; 49 >>>CrossProduct(A,B); Vector([5, 5, -7]) >>>A.(CrossProduct(B,C)); 6 >>>CrossProduct(A,CrossProduct(B,C)); Vector([-267, 204, -3])</lang>
Mathematica
<lang Mathematica>a={3,4,5}; b={4,3,5}; c={-5,-12,-13}; a.b Cross[a,b] a.Cross[b,c] Cross[a,Cross[b,c]]</lang> Output
49
{5,5,-7}
6
{-267,204,-3}MATLAB / Octave
Matlab / Octave use double precesion numbers per default, and pi is a builtin constant value. Arbitrary precision is only implemented in some additional toolboxes (e.g. symbolic toolbox). <lang MATLAB>% Create a named function/subroutine/method to compute the dot product of two vectors.
dot(a,b)
% Create a function to compute the cross product of two vectors.
cross(a,b)
% Optionally create a function to compute the scalar triple product of three vectors.
dot(a,cross(b,c))
% Optionally create a function to compute the vector triple product of three vectors.
cross(a,cross(b,c))
% Compute and display: a • b
cross(a,b)
% Compute and display: a x b
cross(a,b)
% Compute and display: a • b x c, the scaler triple product.
dot(a,cross(b,c))
% Compute and display: a x b x c, the vector triple product.
cross(a,cross(b,c))</lang>
Code for testing:
A = [ 3.0, 4.0, 5.0] B = [ 4.0, 3.0, 5.0] C = [-5.0, -12.0, -13.0] dot(A,B) cross(A,B) dot(A,cross(B,C)) cross(A,cross(B,C))
Output:
>> A = [ 3.0, 4.0, 5.0] >> B = [ 4.0, 3.0, 5.0] >> C = [-5.0, -12.0, -13.0] >> dot(A,B) ans = 49 >> cross(A,B) ans = 5 5 -7 >> dot(A,cross(B,C)) ans = 6 >> cross(A,cross(B,C)) ans = -267 204 -3
Mercury
<lang>:- module vector_product.
- - interface.
- - import_module io.
- - pred main(io::di, io::uo) is det.
- - implementation.
- - import_module int, list, string.
main(!IO) :-
A = vector3d(3, 4, 5),
B = vector3d(4, 3, 5),
C = vector3d(-5, -12, -13),
io.format("A . B = %d\n", [i(A `dot_product` B)], !IO),
io.format("A x B = %s\n", [s(to_string(A `cross_product` B))], !IO),
io.format("A . (B x C) = %d\n", [i(scalar_triple_product(A, B, C))], !IO),
io.format("A x (B x C) = %s\n", [s(to_string(vector_triple_product(A, B, C)))], !IO).
- - type vector3d ---> vector3d(int, int, int).
- - func dot_product(vector3d, vector3d) = int.
dot_product(vector3d(A1, A2, A3), vector3d(B1, B2, B3)) =
A1 * B1 + A2 * B2 + A3 * B3.
- - func cross_product(vector3d, vector3d) = vector3d.
cross_product(vector3d(A1, A2, A3), vector3d(B1, B2, B3)) =
vector3d(A2 * B3 - A3 * B2, A3 * B1 - A1 * B3, A1 * B2 - A2 * B1).
- - func scalar_triple_product(vector3d, vector3d, vector3d) = int.
scalar_triple_product(A, B, C) = A `dot_product` (B `cross_product` C).
- - func vector_triple_product(vector3d, vector3d, vector3d) = vector3d.
vector_triple_product(A, B, C) = A `cross_product` (B `cross_product` C).
- - func to_string(vector3d) = string.
to_string(vector3d(X, Y, Z)) =
string.format("(%d, %d, %d)", [i(X), i(Y), i(Z)]).</lang>
MiniScript
<lang MiniScript>vectorA = [3, 4, 5] vectorB = [4, 3, 5] vectorC = [-5, -12, -13]
dotProduct = function(x, y)
return x[0]*y[0] + x[1]*y[1] + x[2]*y[2]
end function
crossProduct = function(x, y)
return [x[1]*y[2] - x[2]*y[1], x[2]*y[0] - x[0]*y[2], x[0]*y[1] - x[1]*y[0]]
end function
print "Dot Product = " + dotProduct(vectorA, vectorB) print "Cross Product = " + crossProduct(vectorA, vectorB) print "Scalar Triple Product = " + dotProduct(vectorA, crossProduct(vectorB,vectorC)) print "Vector Triple Product = " + crossProduct(vectorA, crossProduct(vectorB,vectorC)) </lang>
- Output:
Dot Product = 49 Cross Product = [5, 5, -7] Scalar Triple Product = 6 Vector Triple Product = [-267, 204, -3]
МК-61/52
<lang>ПП 54 С/П ПП 66 С/П ИП0 ИП3 ИП6 П3 -> П0 -> П6 ИП1 ИП4 ИП7 П4 -> П1 -> П7 ИП2 ИП5 ИП8 П5 -> П2 -> П8 ПП 66 ИП6 ИП7 ИП8 П2 -> П1 -> П0 ИП9 ИПA ИПB П5 -> П4 -> П3 ПП 54 С/П ПП 66 С/П ИП0 ИП3 * ИП1 ИП4 * + ИП2 ИП5 * + В/О ИП1 ИП5 * ИП2 ИП4 * - П9 ИП2 ИП3 * ИП0 ИП5 * - ПA ИП0 ИП4 * ИП1 ИП3 * - ПB В/О</lang>
Instruction: Р0 - a1, Р1 - a2, Р2 - a3, Р3 - b1, Р4 - b2, Р5 - b3, Р6 - c1, Р7 - c2, Р8 - c3; В/О С/П.
Modula-2
<lang modula2>MODULE VectorProducts; FROM RealStr IMPORT RealToStr; FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
PROCEDURE WriteReal(r : REAL); VAR buf : ARRAY[0..31] OF CHAR; BEGIN
RealToStr(r, buf); WriteString(buf)
END WriteReal;
TYPE Vector = RECORD
a,b,c : REAL;
END;
PROCEDURE Dot(u,v : Vector) : REAL; BEGIN
RETURN u.a * v.a
+ u.b * v.b
+ u.c * v.c
END Dot;
PROCEDURE Cross(u,v : Vector) : Vector; BEGIN
RETURN Vector{
u.b*v.c - u.c*v.b,
u.c*v.a - u.a*v.c,
u.a*v.b - u.b*v.a
}
END Cross;
PROCEDURE ScalarTriple(u,v,w : Vector) : REAL; BEGIN
RETURN Dot(u, Cross(v, w))
END ScalarTriple;
PROCEDURE VectorTriple(u,v,w : Vector) : Vector; BEGIN
RETURN Cross(u, Cross(v, w))
END VectorTriple;
PROCEDURE WriteVector(v : Vector); BEGIN
WriteString("<");
WriteReal(v.a);
WriteString(", ");
WriteReal(v.b);
WriteString(", ");
WriteReal(v.c);
WriteString(">")
END WriteVector;
VAR a,b,c : Vector; BEGIN
a := Vector{3.0, 4.0, 5.0};
b := Vector{4.0, 3.0, 5.0};
c := Vector{-5.0, -12.0, -13.0};
WriteVector(a);
WriteString(" dot ");
WriteVector(b);
WriteString(" = ");
WriteReal(Dot(a,b));
WriteLn;
WriteVector(a);
WriteString(" cross ");
WriteVector(b);
WriteString(" = ");
WriteVector(Cross(a,b));
WriteLn;
WriteVector(a);
WriteString(" cross (");
WriteVector(b);
WriteString(" cross ");
WriteVector(c);
WriteString(") = ");
WriteVector(VectorTriple(a,b,c));
WriteLn;
ReadChar
END VectorProducts.</lang>
Nemerle
<lang Nemerle>using System.Console;
module VectorProducts3d {
Dot(x : int * int * int, y : int * int * int) : int
{
def (x1, x2, x3) = x;
def (y1, y2, y3) = y;
(x1 * y1) + (x2 * y2) + (x3 * y3)
}
Cross(x : int * int * int, y : int * int * int) : int * int * int
{
def (x1, x2, x3) = x;
def (y1, y2, y3) = y;
((x2 * y3 - x3 * y2), (x3 * y1 - x1 * y3), (x1 * y2 - x2 * y1))
}
ScalarTriple(a : int * int * int, b : int * int * int, c : int * int * int) : int
{
Dot(a, Cross(b, c))
}
VectorTriple(a : int * int * int, b : int * int * int, c : int * int * int) : int * int * int
{
Cross(a, Cross(b, c))
}
Main() : void
{
def a = (3, 4, 5); def b = (4, 3, 5); def c = (-5, -12, -13);
WriteLine(Dot(a, b)); WriteLine(Cross(a, b));
WriteLine(ScalarTriple(a, b, c));
WriteLine(VectorTriple(a, b, c));
}
}</lang> Outputs
49 (5, 5, -7) 6 (-267, 204, -3)
Never
<lang fsharp>func printv(a[d] : float) -> int {
prints("[" + a[0] + ", " + a[1] + ", " + a[2] + "]\n");
0
}
func dot(a[d1] : float, b[d2] : float) -> float {
a[0] * b[0] + a[1] * b[1] + a[2] * b[2]
}
func cross(a[d1] : float, b[d2] : float) -> [_] : float {
[ a[1] * b[2] - a[2] * b[1], a[2] * b[0] - a[0] * b[2], a[0] * b[1] - a[1] * b[0] ] : float
}
func scalar_triple(a[d1] : float, b[d2] : float, c[d3] : float) -> float {
dot(a, cross(b, c))
}
func vector_triple(a[d1] : float, b[d2] : float, c[d3] : float) -> [_] : float {
cross(a, cross(b, c))
}
func main() -> int {
var a = [ 3.0, 4.0, 5.0 ] : float; var b = [ 4.0, 3.0, 5.0 ] : float; var c = [ -5.0, -12.0, -13.0 ] : float;
printv(a); printv(b); printv(c); printf(dot(a, b)); printv(cross(a, b)); printf(scalar_triple(a, b, c)); printv(vector_triple(a, b, c)); 0
} </lang> Output:
[3.00, 4.00, 5.00] [4.00, 3.00, 5.00] [-5.00, -12.00, -13.00] 49.00 [5.00, 5.00, -7.00] 6.00 [-267.00, 204.00, -3.00]
Nim
<lang nim>import strutils
type Vector3 = array[1..3, float]
proc `$`(a: Vector3): string =
result = "["
for i, x in a:
if i > a.low:
result.add ", "
result.add formatFloat(x, precision = 0)
result.add "]"
proc `~⨯`(a, b: Vector3): Vector3 =
result = [a[2]*b[3] - a[3]*b[2], a[3]*b[1] - a[1]*b[3], a[1]*b[2] - a[2]*b[1]]
proc `~•`[T](a, b: T): float =
for i in a.low..a.high: result += a[i] * b[i]
proc scalartrip(a, b, c: Vector3): float = a ~• (b ~⨯ c)
proc vectortrip(a, b, c: Vector3): Vector3 = a ~⨯ (b ~⨯ c)
let
a = [3.0, 4.0, 5.0] b = [4.0, 3.0, 5.0] c = [-5.0, -12.0, -13.0]
echo "a ⨯ b = ", a ~⨯ b echo "a • b = ", (a ~• b).formatFloat(precision = 0) echo "a . (b ⨯ c) = ", (scalartrip(a, b, c)).formatFloat(precision = 0) echo "a ⨯ (b ⨯ c) = ", vectortrip(a, b, c)</lang> Output:
a ⨯ b = [5, 5, -7] a • b = 49 a . (b ⨯ c) = 6 a ⨯ (b ⨯ c) = [-267, 204, -3]
Objeck
<lang objeck>bundle Default {
class VectorProduct {
function : Main(args : String[]) ~ Nil {
a := Vector3D->New(3.0, 4.0, 5.0);
b := Vector3D->New(4.0, 3.0, 5.0);
c := Vector3D->New(-5.0, -12.0, -13.0);
a->Dot(b)->Print();
a->Cross(b)->Print();
a->ScaleTrip(b, c)->Print();
a->VectorTrip(b, c)->Print();
}
}
class Vector3D {
@a : Float;
@b : Float;
@c : Float;
New(a : Float, b : Float, c : Float) {
@a := a;
@b := b;
@c := c;
}
method : GetA() ~ Float {
return @a;
}
method : GetB() ~ Float {
return @b;
}
method : GetC() ~ Float {
return @c;
}
method : public : Dot(vec : Vector3D) ~ Float {
return @a * vec->GetA() + @b * vec->GetB() + @c * vec->GetC();
}
method : public : Cross(vec : Vector3D) ~ Vector3D {
newA := @b * vec->GetC() - @c * vec->GetB();
newB := @c * vec->GetA() - @a * vec->GetC();
newC := @a * vec->GetB() - @b * vec->GetA();
return Vector3D->New(newA, newB, newC); }
method : public : ScaleTrip(vec_b: Vector3D, vec_c : Vector3D) ~ Float {
return Dot(vec_b->Cross(vec_c));
}
method : public : Print() ~ Nil {
IO.Console->Print('<')->Print(@a)->Print(" ,")
->Print(@b)->Print(", ")->Print(@c)->PrintLine('>');
}
method : public : VectorTrip(vec_b: Vector3D, vec_c : Vector3D) ~ Vector3D {
return Cross(vec_b->Cross(vec_c));
}
}
} </lang>
Output:
49<5 ,5, -7> 6<-267 ,204, -3>
OCaml
<lang ocaml>let a = (3.0, 4.0, 5.0) let b = (4.0, 3.0, 5.0) let c = (-5.0, -12.0, -13.0)
let string_of_vector (x,y,z) =
Printf.sprintf "(%g, %g, %g)" x y z
let dot (a1, a2, a3) (b1, b2, b3) =
(a1 *. b1) +. (a2 *. b2) +. (a3 *. b3)
let cross (a1, a2, a3) (b1, b2, b3) =
(a2 *. b3 -. a3 *. b2, a3 *. b1 -. a1 *. b3, a1 *. b2 -. a2 *. b1)
let scalar_triple a b c =
dot a (cross b c)
let vector_triple a b c =
cross a (cross b c)
let () =
Printf.printf "a: %s\n" (string_of_vector a); Printf.printf "b: %s\n" (string_of_vector b); Printf.printf "c: %s\n" (string_of_vector c); Printf.printf "a . b = %g\n" (dot a b); Printf.printf "a x b = %s\n" (string_of_vector (cross a b)); Printf.printf "a . (b x c) = %g\n" (scalar_triple a b c); Printf.printf "a x (b x c) = %s\n" (string_of_vector (vector_triple a b c));
- </lang>
outputs:
a: (3, 4, 5) b: (4, 3, 5) c: (-5, -12, -13) a . b = 49 a x b = (5, 5, -7) a . (b x c) = 6 a x (b x c) = (-267, 204, -3)
Octave
Octave handles naturally vectors / matrices. <lang octave>a = [3, 4, 5]; b = [4, 3, 5]; c = [-5, -12, -13];
function r = s3prod(a, b, c)
r = dot(a, cross(b, c));
endfunction
function r = v3prod(a, b, c)
r = cross(a, cross(b, c));
endfunction
% 49 dot(a, b) % or matrix-multiplication between row and column vectors a * b'
% 5 5 -7 cross(a, b) % only for 3d-vectors
% 6 s3prod(a, b, c)
% -267 204 -3 v3prod(a, b, c)</lang>
ooRexx
<lang ooRexx> a = .vector~new(3, 4, 5); b = .vector~new(4, 3, 5); c = .vector~new(-5, -12, -13);
say a~dot(b) say a~cross(b) say a~scalarTriple(b, c) say a~vectorTriple(b, c)
- class vector
- method init
expose x y z use arg x, y, z
- attribute x get
- attribute y get
- attribute z get
-- dot product operation
- method dot
expose x y z use strict arg other
return x * other~x + y * other~y + z * other~z
-- cross product operation
- method cross
expose x y z use strict arg other
newX = y * other~z - z * other~y newY = z * other~x - x * other~z newZ = x * other~y - y * other~x return self~class~new(newX, newY, newZ)
-- scalar triple product
- method scalarTriple
use strict arg vectorB, vectorC return self~dot(vectorB~cross(vectorC))
-- vector triple product
- method vectorTriple
use strict arg vectorB, vectorC return self~cross(vectorB~cross(vectorC))
- method string
expose x y z return "<"||x", "y", "z">"
</lang> Output:
49 <5, 5, -7> 6 <-267, 204, -3>
PARI/GP
<lang parigp>dot(u,v)={
sum(i=1,#u,u[i]*v[i])
}; cross(u,v)={
[u[2]*v[3] - u[3]*v[2], u[3]*v[1] - u[1]*v[3], u[1]*v[2] - u[2]*v[1]]
}; striple(a,b,c)={
dot(a,cross(b,c))
}; vtriple(a,b,c)={
cross(a,cross(b,c))
};
a = [3,4,5]; b = [4,3,5]; c = [-5,-12,-13]; dot(a,b) cross(a,b) striple(a,b,c) vtriple(a,b,c)</lang> Output:
49 [5, 5, -7] 6 [-267, 204, -3]
Pascal
<lang pascal>Program VectorProduct (output);
type
Tvector = record x, y, z: double end;
function dotProduct(a, b: Tvector): double; begin
dotProduct := a.x*b.x + a.y*b.y + a.z*b.z;
end;
function crossProduct(a, b: Tvector): Tvector; begin
crossProduct.x := a.y*b.z - a.z*b.y; crossProduct.y := a.z*b.x - a.x*b.z; crossProduct.z := a.x*b.y - a.y*b.x;
end;
function scalarTripleProduct(a, b, c: Tvector): double; begin
scalarTripleProduct := dotProduct(a, crossProduct(b, c));
end;
function vectorTripleProduct(a, b, c: Tvector): Tvector; begin
vectorTripleProduct := crossProduct(a, crossProduct(b, c));
end;
procedure printVector(a: Tvector); begin
writeln(a.x:15:8, a.y:15:8, a.z:15:8);
end;
var
a: Tvector = (x: 3; y: 4; z: 5); b: Tvector = (x: 4; y: 3; z: 5); c: Tvector = (x:-5; y:-12; z:-13);
begin
write('a: '); printVector(a);
write('b: '); printVector(b);
write('c: '); printVector(c);
writeln('a . b: ', dotProduct(a,b):15:8);
write('a x b: '); printVector(crossProduct(a,b));
writeln('a . (b x c): ', scalarTripleProduct(a,b,c):15:8);
write('a x (b x c): '); printVector(vectorTripleProduct(a,b,c));
end.</lang> Output:
a: 3.00000000 4.00000000 5.00000000 b: 4.00000000 3.00000000 5.00000000 c: -5.00000000 -12.00000000 -13.00000000 a . b: 49.00000000 a x b: 5.00000000 5.00000000 -7.00000000 a . (b x c): 6.00000000 a x (b x c): -267.00000000 204.00000000 -3.00000000
Perl
<lang Perl>package Vector; use List::Util 'sum'; use List::MoreUtils 'pairwise';
sub new { shift; bless [@_] }
use overload (
'""' => sub { "(@{+shift})" },
'&' => sub { sum pairwise { $a * $b } @{+shift}, @{+shift} },
'^' => sub {
my @a = @{+shift};
my @b = @{+shift};
bless [ $a[1]*$b[2] - $a[2]*$b[1],
$a[2]*$b[0] - $a[0]*$b[2],
$a[0]*$b[1] - $a[1]*$b[0] ]
},
);
package main; my $a = Vector->new(3, 4, 5); my $b = Vector->new(4, 3, 5); my $c = Vector->new(-5, -12, -13);
print "a = $a b = $b c = $c\n"; print "$a . $b = ", $a & $b, "\n"; print "$a x $b = ", $a ^ $b, "\n"; print "$a . ($b x $c) = ", $a & ($b ^ $c), "\n"; print "$a x ($b x $c) = ", $a ^ ($b ^ $c), "\n";</lang>
Output:
a = (3 4 5) b = (4 3 5) c = (-5 -12 -13) (3 4 5) . (4 3 5) = 49 (3 4 5) x (4 3 5) = (5 5 -7) (3 4 5) . ((4 3 5) x (-5 -12 -13)) = 6 (3 4 5) x ((4 3 5) x (-5 -12 -13)) = (-267 204 -3)
Phix
<lang Phix>function dot_product(sequence a, b)
return sum(sq_mul(a,b))
end function
function cross_product(sequence a, b) integer {a1,a2,a3} = a, {b1,b2,b3} = b
return {a2*b3-a3*b2, a3*b1-a1*b3, a1*b2-a2*b1}
end function
function scalar_triple_product(sequence a, b, c)
return dot_product(a,cross_product(b,c))
end function
function vector_triple_product(sequence a, b, c)
return cross_product(a,cross_product(b,c))
end function
constant a = {3, 4, 5}, b = {4, 3, 5}, c = {-5, -12, -13}
puts(1," a . b = ") ?dot_product(a,b) puts(1," a x b = ") ?cross_product(a,b) puts(1,"a . (b x c) = ") ?scalar_triple_product(a,b,c) puts(1,"a x (b x c) = ") ?vector_triple_product(a,b,c)</lang>
- Output:
a . b = 49
a x b = {5,5,-7}
a . (b x c) = 6
a x (b x c) = {-267,204,-3}
PHP
<lang PHP><?php
class Vector { private $values;
public function setValues(array $values) { if (count($values) != 3) throw new Exception('Values must contain exactly 3 values'); foreach ($values as $value) if (!is_int($value) && !is_float($value)) throw new Exception('Value "' . $value . '" has an invalid type'); $this->values = $values; }
public function getValues() { if ($this->values == null) $this->setValues(array ( 0, 0, 0 )); return $this->values; }
public function Vector(array $values) { $this->setValues($values); }
public static function dotProduct(Vector $va, Vector $vb) { $a = $va->getValues(); $b = $vb->getValues(); return ($a[0] * $b[0]) + ($a[1] * $b[1]) + ($a[2] * $b[2]); }
public static function crossProduct(Vector $va, Vector $vb) { $a = $va->getValues(); $b = $vb->getValues(); return new Vector(array ( ($a[1] * $b[2]) - ($a[2] * $b[1]), ($a[2] * $b[0]) - ($a[0] * $b[2]), ($a[0] * $b[1]) - ($a[1] * $b[0]) )); }
public static function scalarTripleProduct(Vector $va, Vector $vb, Vector $vc) { return self::dotProduct($va, self::crossProduct($vb, $vc)); }
public static function vectorTrippleProduct(Vector $va, Vector $vb, Vector $vc) { return self::crossProduct($va, self::crossProduct($vb, $vc)); } }
class Program {
public function Program() { $a = array ( 3, 4, 5 ); $b = array ( 4, 3, 5 ); $c = array ( -5, -12, -13 ); $va = new Vector($a); $vb = new Vector($b); $vc = new Vector($c);
$result1 = Vector::dotProduct($va, $vb); $result2 = Vector::crossProduct($va, $vb)->getValues(); $result3 = Vector::scalarTripleProduct($va, $vb, $vc); $result4 = Vector::vectorTrippleProduct($va, $vb, $vc)->getValues();
printf("\n"); printf("A = (%0.2f, %0.2f, %0.2f)\n", $a[0], $a[1], $a[2]); printf("B = (%0.2f, %0.2f, %0.2f)\n", $b[0], $b[1], $b[2]); printf("C = (%0.2f, %0.2f, %0.2f)\n", $c[0], $c[1], $c[2]); printf("\n"); printf("A · B = %0.2f\n", $result1); printf("A × B = (%0.2f, %0.2f, %0.2f)\n", $result2[0], $result2[1], $result2[2]); printf("A · (B × C) = %0.2f\n", $result3); printf("A × (B × C) =(%0.2f, %0.2f, %0.2f)\n", $result4[0], $result4[1], $result4[2]); } }
new Program(); ?> </lang>
Output:
A = (3.00, 4.00, 5.00) B = (4.00, 3.00, 5.00) C = (-5.00, -12.00, -13.00) A · B = 49.00 A × B = (5.00, 5.00, -7.00) A · (B × C) = 6.00 A × (B × C) =(-267.00, 204.00, -3.00)
PicoLisp
<lang PicoLisp>(de dotProduct (A B)
(sum * A B) )
(de crossProduct (A B)
(list
(- (* (cadr A) (caddr B)) (* (caddr A) (cadr B)))
(- (* (caddr A) (car B)) (* (car A) (caddr B)))
(- (* (car A) (cadr B)) (* (cadr A) (car B))) ) )
(de scalarTriple (A B C)
(dotProduct A (crossProduct B C)) )
(de vectorTriple (A B C)
(crossProduct A (crossProduct B C)) )</lang>
Test:
(setq A ( 3 4 5) B ( 4 3 5) C (-5 -12 -13) ) : (dotProduct A B) -> 49 : (crossProduct A B) -> (5 5 -7) : (scalarTriple A B C) -> 6 : (vectorTriple A B C) -> (-267 204 -3)
PL/I
<lang PL/I>/* dot product, cross product, etc. 4 June 2011 */
test_products: procedure options (main);
declare a(3) fixed initial (3, 4, 5); declare b(3) fixed initial (4, 3, 5); declare c(3) fixed initial (-5, -12, -13); declare e(3) fixed;
put skip list ('a . b =', dot_product(a, b));
call cross_product(a, b, e); put skip list ('a x b =', e);
put skip list ('a . (b x c) =', scalar_triple_product(a, b, c));
call vector_triple_product(a, b, c, e); put skip list ('a x (b x c) =', e);
dot_product: procedure (a, b) returns (fixed);
declare (a, b) (*) fixed; return (sum(a*b));
end dot_product;
cross_product: procedure (a, b, c);
declare (a, b, c) (*) fixed; c(1) = a(2)*b(3) - a(3)*b(2); c(2) = a(3)*b(1) - a(1)*b(3); c(3) = a(1)*b(2) - a(2)*b(1);
end cross_product;
scalar_triple_product: procedure (a, b, c) returns (fixed);
declare (a, b, c)(*) fixed; declare t(hbound(a, 1)) fixed; call cross_product(b, c, t); return (dot_product(a, t));
end scalar_triple_product;
vector_triple_product: procedure (a, b, c, e);
declare (a, b, c, e)(*) fixed; declare t(hbound(a,1)) fixed; call cross_product(b, c, t); call cross_product(a, t, e);
end vector_triple_product;
end test_products;</lang> Results:
a . b = 49 a x b = 5 5 -7 a . (b x c) = 6 a x (b x c) = -267 204 -3
<lang PL/I>/* This version uses the ability of PL/I to return arrays. */
/* dot product, cross product, etc. 6 June 2011 */
test_products: procedure options (main);
define structure 1 vector, 2 vec(3) fixed; declare (a, b, c) type(vector);
a.vec(1) = 3; a.vec(2) = 4; a.vec(3) = 5; b.vec(1) = 4; b.vec(2) = 3; b.vec(3) = 5; c.vec(1) = -5; c.vec(2) = -12; c.vec(3) = -13;
put skip list ('a . b =', dot_product (a, b) );
put skip list ('a x b =', cross_product(a, b).vec);
put skip list ('a . (b x c) =', scalar_triple_product(a, b, c) );
put skip list ('a x (b x c) =', vector_triple_product(a, b, c).vec);
dot_product: procedure (a, b) returns (fixed);
declare (a, b) type(vector); return (sum(a.vec*b.vec));
end dot_product;
cross_product: procedure (a, b) returns (type(vector));
declare (a, b) type(vector); declare c type vector; c.vec(1) = a.vec(2)*b.vec(3) - a.vec(3)*b.vec(2); c.vec(2) = a.vec(3)*b.vec(1) - a.vec(1)*b.vec(3); c.vec(3) = a.vec(1)*b.vec(2) - a.vec(2)*b.vec(1); return (c);
end cross_product;
scalar_triple_product: procedure (a, b, c) returns (fixed);
declare (a, b, c) type(vector); declare t type (vector); t = cross_product(b, c); return (dot_product(a, t));
end scalar_triple_product;
vector_triple_product: procedure (a, b, c) returns (type(vector));
declare (a, b, c) type(vector); declare (t, e) type (vector); t = cross_product(b, c); e = cross_product(a, t); return (e);
end vector_triple_product;
end test_products;</lang> The output is:
a . b = 49 a x b = 5 5 -7 a . (b x c) = 6 a x (b x c) = -267 204 -3
PowerShell
<lang PowerShell> function dot-product($a,$b) {
$a[0]*$b[0] + $a[1]*$b[1] + $a[2]*$b[2]
}
function cross-product($a,$b) {
$v1 = $a[1]*$b[2] - $a[2]*$b[1] $v2 = $a[2]*$b[0] - $a[0]*$b[2] $v3 = $a[0]*$b[1] - $a[1]*$b[0] @($v1,$v2,$v3)
}
function scalar-triple-product($a,$b,$c) {
dot-product $a (cross-product $b $c)
}
function vector-triple-product($a,$b) {
cross-product $a (cross-product $b $c)
}
$a = @(3, 4, 5) $b = @(4, 3, 5) $c = @(-5, -12, -13)
"a.b = $(dot-product $a $b)" "axb = $(cross-product $a $b)" "a.(bxc) = $(scalar-triple-product $a $b $c)" "ax(bxc) = $(vector-triple-product $a $b $c)" </lang> Output:
a.b = 49 axb = 5 5 -7 a.(bxc) = 6 ax(bxc) = -267 204 -3
Prolog
Works with SWI-Prolog. <lang prolog> dot_product([A1, A2, A3], [B1, B2, B3], Ans) :-
Ans is A1 * B1 + A2 * B2 + A3 * B3.
cross_product([A1, A2, A3], [B1, B2, B3], Ans) :-
T1 is A2 * B3 - A3 * B2, T2 is A3 * B1 - A1 * B3, T3 is A1 * B2 - A2 * B1, Ans = [T1, T2, T3].
scala_triple(A, B, C, Ans) :-
cross_product(B, C, Temp), dot_product(A, Temp, Ans).
vector_triple(A, B, C, Ans) :-
cross_product(B, C, Temp), cross_product(A, Temp, Ans).
</lang> Output:
?- dot_product([3.0, 4.0, 5.0], [4.0, 3.0, 5.0], Ans). Ans = 49.0. ?- cross_product([3.0, 4.0, 5.0], [4.0, 3.0, 5.0], Ans). Ans = [5.0, 5.0, -7.0]. ?- scala_triple([3.0, 4.0, 5.0], [4.0, 3.0, 5.0], [-5.0, -12.0, -13.0], Ans). Ans = 6.0. ?- vector_triple([3.0, 4.0, 5.0], [4.0, 3.0, 5.0], [-5.0, -12.0, -13.0], Ans). Ans = [-267.0, 204.0, -3.0].
PureBasic
<lang PureBasic>Structure vector
x.f y.f z.f
EndStructure
- convert vector to a string for display
Procedure.s toString(*v.vector)
ProcedureReturn "[" + StrF(*v\x, 2) + ", " + StrF(*v\y, 2) + ", " + StrF(*v\z, 2) + "]"
EndProcedure
Procedure.f dotProduct(*a.vector, *b.vector)
ProcedureReturn *a\x * *b\x + *a\y * *b\y + *a\z * *b\z
EndProcedure
Procedure crossProduct(*a.vector, *b.vector, *r.vector)
*r\x = *a\y * *b\z - *a\z * *b\y *r\y = *a\z * *b\x - *a\x * *b\z *r\z = *a\x * *b\y - *a\y * *b\x
EndProcedure
Procedure.f scalarTriple(*a.vector, *b.vector, *c.vector)
Protected r.vector crossProduct(*b, *c, r) ProcedureReturn dotProduct(*a, r)
EndProcedure
Procedure vectorTriple(*a.vector, *b.vector, *c.vector, *r.vector)
Protected r.vector crossProduct(*b, *c, r) crossProduct(*a, r, *r)
EndProcedure
If OpenConsole()
Define.vector a, b, c, r
a\x = 3: a\y = 4: a\z = 5
b\x = 4: b\y = 3: b\z = 5
c\x = -5: c\y = -12: c\z = -13
PrintN("a = " + toString(a) + ", b = " + toString(b) + ", c = " + toString(c))
PrintN("a . b = " + StrF(dotProduct(a, b), 2))
crossProduct(a, b, r)
PrintN("a x b = " + toString(r))
PrintN("a . b x c = " + StrF(scalarTriple(a, b, c), 2))
vectorTriple(a, b, c, r)
PrintN("a x b x c = " + toString(r))
Print(#CRLF$ + #CRLF$ + "Press ENTER to exit"): Input()
CloseConsole()
EndIf</lang> Sample output:
a = [3.00, 4.00, 5.00], b = [4.00, 3.00, 5.00], c = [-5.00, -12.00, -13.00] a . b = 49.00 a x b = [5.00, 5.00, -7.00] a . b x c = 6.00 a x b x c = [-267.00, 204.00, -3.00]
Python
The solution is in the form of an Executable library. <lang python>def crossp(a, b):
Cross product of two 3D vectors assert len(a) == len(b) == 3, 'For 3D vectors only' a1, a2, a3 = a b1, b2, b3 = b return (a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1)
def dotp(a,b):
Dot product of two eqi-dimensioned vectors assert len(a) == len(b), 'Vector sizes must match' return sum(aterm * bterm for aterm,bterm in zip(a, b))
def scalartriplep(a, b, c):
Scalar triple product of three vectors: "a . (b x c)" return dotp(a, crossp(b, c))
def vectortriplep(a, b, c):
Vector triple product of three vectors: "a x (b x c)" return crossp(a, crossp(b, c))
if __name__ == '__main__':
a, b, c = (3, 4, 5), (4, 3, 5), (-5, -12, -13)
print("a = %r; b = %r; c = %r" % (a, b, c))
print("a . b = %r" % dotp(a,b))
print("a x b = %r" % (crossp(a,b),))
print("a . (b x c) = %r" % scalartriplep(a, b, c))
print("a x (b x c) = %r" % (vectortriplep(a, b, c),))</lang>
- Output:
a = (3, 4, 5); b = (4, 3, 5); c = (-5, -12, -13) a . b = 49 a x b = (5, 5, -7) a . (b x c) = 6 a x (b x c) = (-267, 204, -3)
- Note
The popular numpy package has functions for dot and cross products.
R
<lang rsplus>#===============================================================
- Vector products
- R implementation
- ===============================================================
a <- c(3, 4, 5) b <- c(4, 3, 5) c <- c(-5, -12, -13)
- ---------------------------------------------------------------
- Dot product
- ---------------------------------------------------------------
dotp <- function(x, y) {
if (length(x) == length(y)) {
sum(x*y)
}
}
- ---------------------------------------------------------------
- Cross product
- ---------------------------------------------------------------
crossp <- function(x, y) {
if (length(x) == 3 && length(y) == 3) {
c(x[2]*y[3] - x[3]*y[2], x[3]*y[1] - x[1]*y[3], x[1]*y[2] - x[2]*y[1])
}
}
- ---------------------------------------------------------------
- Scalar triple product
- ---------------------------------------------------------------
scalartriplep <- function(x, y, z) {
if (length(x) == 3 && length(y) == 3 && length(z) == 3) {
dotp(x, crossp(y, z))
}
}
- ---------------------------------------------------------------
- Vector triple product
- ---------------------------------------------------------------
vectortriplep <- function(x, y, z) {
if (length(x) == 3 && length(y) == 3 && length(z) == 3) {
crosssp(x, crossp(y, z))
}
}
- ---------------------------------------------------------------
- Compute and print
- ---------------------------------------------------------------
cat("a . b =", dotp(a, b)) cat("a x b =", crossp(a, b)) cat("a . (b x c) =", scalartriplep(a, b, c)) cat("a x (b x c) =", vectortriplep(a, b, c))</lang>
- Output:
a . b = 49 a x b = 5 5 -7 a . (b x c) = 6 a x (b x c) = -267 204 -3
Note: R has built-in functions for vector and matrix multiplications. Examples: "crossprod", %*% for inner and %o% for outer product.
Racket
<lang Racket>
- lang racket
(define (dot-product X Y)
(for/sum ([x (in-vector X)] [y (in-vector Y)]) (* x y)))
(define (cross-product X Y)
(define len (vector-length X)) (for/vector ([n len]) (define (ref V i) (vector-ref V (modulo (+ n i) len))) (- (* (ref X 1) (ref Y 2)) (* (ref X 2) (ref Y 1)))))
(define (scalar-triple-product X Y Z)
(dot-product X (cross-product Y Z)))
(define (vector-triple-product X Y Z)
(cross-product X (cross-product Y Z)))
(define A '#(3 4 5)) (define B '#(4 3 5)) (define C '#(-5 -12 -13))
(printf "A = ~s\n" A) (printf "B = ~s\n" B) (printf "C = ~s\n" C) (newline)
(printf "A . B = ~s\n" (dot-product A B)) (printf "A x B = ~s\n" (cross-product A B)) (printf "A . B x C = ~s\n" (scalar-triple-product A B C)) (printf "A x B x C = ~s\n" (vector-triple-product A B C)) </lang>
Raku
(formerly Perl 6)
<lang perl6>sub infix:<⋅> { [+] @^a »*« @^b }
sub infix:<⨯>([$a1, $a2, $a3], [$b1, $b2, $b3]) {
[ $a2*$b3 - $a3*$b2,
$a3*$b1 - $a1*$b3,
$a1*$b2 - $a2*$b1 ];
}
sub scalar-triple-product { @^a ⋅ (@^b ⨯ @^c) } sub vector-triple-product { @^a ⨯ (@^b ⨯ @^c) }
my @a = <3 4 5>; my @b = <4 3 5>; my @c = <-5 -12 -13>;
say (:@a, :@b, :@c); say "a ⋅ b = { @a ⋅ @b }"; say "a ⨯ b = <{ @a ⨯ @b }>"; say "a ⋅ (b ⨯ c) = { scalar-triple-product(@a, @b, @c) }"; say "a ⨯ (b ⨯ c) = <{ vector-triple-product(@a, @b, @c) }>";</lang>
- Output:
("a" => ["3", "4", "5"], "b" => ["4", "3", "5"], "c" => ["-5", "-12", "-13"])
a ⋅ b = 49
a ⨯ b = <5 5 -7>
a ⋅ (b ⨯ c) = 6
a ⨯ (b ⨯ c) = <-267 204 -3>REXX
<lang rexx>/*REXX program computes the products: dot, cross, scalar triple, and vector triple.*/
a= 3 4 5
b= 4 3 5 /*(positive numbers don't need quotes.)*/
c= "-5 -12 -13"
call tellV 'vector A =', a /*show the A vector, aligned numbers.*/ call tellV 'vector B =', b /* " " B " " " */ call tellV 'vector C =', c /* " " C " " " */ say call tellV ' dot product [A∙B] =', dot(a, b) call tellV 'cross product [AxB] =', cross(a, b) call tellV 'scalar triple product [A∙(BxC)] =', dot(a, cross(b, c) ) call tellV 'vector triple product [Ax(BxC)] =', cross(a, cross(b, c) ) exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ cross: procedure; arg $1 $2 $3,@1 @2 @3; return $2*@3 -$3*@2 $3*@1 -$1*@3 $1*@2 -$2*@1 dot: procedure; arg $1 $2 $3,@1 @2 @3; return $1*@1 + $2*@2 + $3*@3 /*──────────────────────────────────────────────────────────────────────────────────────*/ tellV: procedure; parse arg name,x y z /*obtain name, values.*/
w=max(4, length(x), length(y), length(z) ) /*max width of numbers*/
say right(name, 40) right(x,w) right(y,w) right(z,w) /*enforce # alignment.*/
return /* [↑] display vector*/</lang>
- output when using the default inputs:
vector A = 3 4 5
vector B = 4 3 5
vector C = -5 -12 -13
dot product [A∙B] = 49
cross product [AxB] = 5 5 -7
scalar triple product [A∙(BxC)] = 6
vector triple product [Ax(BxC)] = -267 204 -3
Ring
<lang ring>
- Project : Vector products
d = list(3) e = list(3) a = [3, 4, 5] b = [4, 3, 5] c = [-5, -12, -13]
see "a . b = " + dot(a,b) + nl cross(a,b,d) see "a x b = (" + d[1] + ", " + d[2] + ", " + d[3] + ")" + nl see "a . (b x c) = " + scalartriple(a,b,c) + nl vectortriple(a,b,c,d)
def dot(a,b)
sum = 0
for n=1 to len(a)
sum = sum + a[n]*b[n]
next
return sum
func cross(a,b,d)
d = [a[2]*b[3]-a[3]*b[2], a[3]*b[1]-a[1]*b[3], a[1]*b[2]-a[2]*b[1]]
func scalartriple(a,b,c)
cross(b,c,d)
return dot(a,d)
func vectortriple(a,b,c,d)
cross(b,c,d)
cross(a,d,e)
see "a x (b x c) = (" + e[1] + ", " +e[2] + ", " + e[3] + ")"
</lang> Output:
a . b = 49 a x b = (5, 5, -7) a . (b x c) = 6 a x (b x c) = (-267, 204, -3)
Ruby
Dot product is also known as inner product. The standard library already defines Vector#inner_product and Vector# cross_product, so this program only defines the other two methods. <lang ruby>require 'matrix'
class Vector
def scalar_triple_product(b, c) self.inner_product(b.cross_product c) end
def vector_triple_product(b, c) self.cross_product(b.cross_product c) end
end
a = Vector[3, 4, 5] b = Vector[4, 3, 5] c = Vector[-5, -12, -13]
puts "a dot b = #{a.inner_product b}" puts "a cross b = #{a.cross_product b}" puts "a dot (b cross c) = #{a.scalar_triple_product b, c}" puts "a cross (b cross c) = #{a.vector_triple_product b, c}"</lang>
Output:
a dot b = 49 a cross b = Vector[5, 5, -7] a dot (b cross c) = 6 a cross (b cross c) = Vector[-267, 204, -3]
Rust
<lang rust>#[derive(Debug)] struct Vector {
x: f64, y: f64, z: f64,
}
impl Vector {
fn new(x: f64, y: f64, z: f64) -> Self {
Vector {
x: x,
y: y,
z: z,
}
}
fn dot_product(&self, other: &Vector) -> f64 {
(self.x * other.x) + (self.y * other.y) + (self.z * other.z)
}
fn cross_product(&self, other: &Vector) -> Vector {
Vector::new(self.y * other.z - self.z * other.y,
self.z * other.x - self.x * other.z,
self.x * other.y - self.y * other.x)
}
fn scalar_triple_product(&self, b: &Vector, c: &Vector) -> f64 {
self.dot_product(&b.cross_product(&c))
}
fn vector_triple_product(&self, b: &Vector, c: &Vector) -> Vector {
self.cross_product(&b.cross_product(&c))
}
}
fn main(){
let a = Vector::new(3.0, 4.0, 5.0); let b = Vector::new(4.0, 3.0, 5.0); let c = Vector::new(-5.0, -12.0, -13.0);
println!("a . b = {}", a.dot_product(&b));
println!("a x b = {:?}", a.cross_product(&b));
println!("a . (b x c) = {}", a.scalar_triple_product(&b, &c));
println!("a x (b x c) = {:?}", a.vector_triple_product(&b, &c));
}</lang>
Output:
a . b = 49
a x b = Vector { x: 5, y: 5, z: -7 }
a . (b x c) = 6
a x (b x c) = Vector { x: -267, y: 204, z: -3 }Scala
<lang scala>case class Vector3D(x:Double, y:Double, z:Double) {
def dot(v:Vector3D):Double=x*v.x + y*v.y + z*v.z; def cross(v:Vector3D)=Vector3D(y*v.z - z*v.y, z*v.x - x*v.z, x*v.y - y*v.x) def scalarTriple(v1:Vector3D, v2:Vector3D)=this dot (v1 cross v2) def vectorTriple(v1:Vector3D, v2:Vector3D)=this cross (v1 cross v2)
}
object VectorTest {
def main(args:Array[String])={
val a=Vector3D(3,4,5)
val b=Vector3D(4,3,5)
val c=Vector3D(-5,-12,-13)
println(" a . b : " + (a dot b))
println(" a x b : " + (a cross b))
println("a . (b x c) : " + (a scalarTriple(b, c)))
println("a x (b x c) : " + (a vectorTriple(b, c)))
}
}</lang>
- Output:
a . b : 49.0
a x b : Vector3D(5.0,5.0,-7.0)
a . (b x c) : 6.0
a x (b x c) : Vector3D(-267.0,204.0,-3.0)Scheme
Using modified dot-product function from the Dot product task. <lang scheme>(define (dot-product A B)
(apply + (map * (vector->list A) (vector->list B))))
(define (cross-product A B) (define len (vector-length A)) (define xp (make-vector (vector-length A) #f)) (let loop ((n 0)) (vector-set! xp n (- (* (vector-ref A (modulo (+ n 1) len)) (vector-ref B (modulo (+ n 2) len))) (* (vector-ref A (modulo (+ n 2) len)) (vector-ref B (modulo (+ n 1) len))))) (if (eqv? len (+ n 1)) xp (loop (+ n 1)))))
(define (scalar-triple-product A B C) (dot-product A (cross-product B C)))
(define (vector-triple-product A B C) (cross-product A (cross-product B C)))
(define A #( 3 4 5))
(define B #(4 3 5))
(define C #(-5 -12 -13))
(display "A = ")(display A)(newline) (display "B = ")(display B)(newline) (display "C = ")(display C)(newline) (newline) (display "A . B = ")(display (dot-product A B))(newline) (display "A x B = ")(display (cross-product A B))(newline) (display "A . B x C = ")(display (scalar-triple-product A B C))(newline) (display "A x B x C = ") (display (vector-triple-product A B C))(newline)</lang>
Output:
A = #(3 4 5) B = #(4 3 5) C = #(-5 -12 -13) A . B = 49 A x B = #(5 5 -7) A . B x C = 6 A x B x C = #(-267 204 -3)
Seed7
The program below uses Seed7s capaibility to define operator symbols. The operators dot and X are defined with with priority 6 and assiciativity left-to-right. <lang seed7>$ include "seed7_05.s7i";
include "float.s7i";
const type: vec3 is new struct
var float: x is 0.0; var float: y is 0.0; var float: z is 0.0; end struct;
const func vec3: vec3 (in float: x, in float: y, in float: z) is func
result var vec3: aVector is vec3.value; begin aVector.x := x; aVector.y := y; aVector.z := z; end func;
$ syntax expr: .(). dot .() is -> 6; const func float: (in vec3: a) dot (in vec3: b) is
return a.x*b.x + a.y*b.y + a.z*b.z;
$ syntax expr: .(). X .() is -> 6; const func vec3: (in vec3: a) X (in vec3: b) is
return vec3(a.y*b.z - a.z*b.y,
a.z*b.x - a.x*b.z,
a.x*b.y - a.y*b.x);
const func string: str (in vec3: v) is
return "(" <& v.x <& ", " <& v.y <& ", " <& v.z <& ")";
enable_output(vec3);
const func float: scalarTriple (in vec3: a, in vec3: b, in vec3: c) is
return a dot (b X c);
const func vec3: vectorTriple (in vec3: a, in vec3: b, in vec3: c) is
return a X (b X c);
const proc: main is func
local
const vec3: a is vec3(3.0, 4.0, 5.0);
const vec3: b is vec3(4.0, 3.0, 5.0);
const vec3: c is vec3(-5.0, -12.0, -13.0);
begin
writeln("a = " <& a <& ", b = " <& b <& ", c = " <& c);
writeln("a . b = " <& a dot b);
writeln("a x b = " <& a X b);
writeln("a .(b x c) = " <& scalarTriple(a, b, c));
writeln("a x(b x c) = " <& vectorTriple(a, b, c));
end func;</lang>
- Output:
a = (3.0, 4.0, 5.0), b = (4.0, 3.0, 5.0), c = (-5.0, -12.0, -13.0) a . b = 49.0 a x b = (5.0, 5.0, -7.0) a .(b x c) = 6.0 a x(b x c) = (-267.0, 204.0, -3.0)
Sidef
<lang ruby>class MyVector(x, y, z) {
method ∙(vec) {
[self{:x,:y,:z}] »*« [vec{:x,:y,:z}] «+»
}
method ⨉(vec) {
MyVector(self.y*vec.z - self.z*vec.y,
self.z*vec.x - self.x*vec.z,
self.x*vec.y - self.y*vec.x)
}
method to_s {
"(#{x}, #{y}, #{z})"
}
} var a = MyVector(3, 4, 5) var b = MyVector(4, 3, 5) var c = MyVector(-5, -12, -13) say "a=#{a}; b=#{b}; c=#{c};" say "a ∙ b = #{a ∙ b}" say "a ⨉ b = #{a ⨉ b}" say "a ∙ (b ⨉ c) = #{a ∙ (b ⨉ c)}" say "a ⨉ (b ⨉ c) = #{a ⨉ (b ⨉ c)}"</lang>
- Output:
a=(3, 4, 5); b=(4, 3, 5); c=(-5, -12, -13); a ∙ b = 49 a ⨉ b = (5, 5, -7) a ∙ (b ⨉ c) = 6 a ⨉ (b ⨉ c) = (-267, 204, -3)
Stata
<lang stata>mata real scalar sprod(real colvector u, real colvector v) { return(u[1]*v[1] + u[2]*v[2] + u[3]*v[3]) }
real colvector vprod(real colvector u, real colvector v) { return(u[2]*v[3]-u[3]*v[2]\u[3]*v[1]-u[1]*v[3]\u[1]*v[2]-u[2]*v[1]) }
real scalar striple(real colvector u, real colvector v, real colvector w) { return(sprod(u, vprod(v, w))) }
real colvector vtriple(real colvector u, real colvector v, real colvector w) { return(vprod(u, vprod(v, w))) }
a = 3\4\5 b = 4\3\5 c = -5\-12\-13
sprod(a, b)
49
vprod(a, b)
1 +------+ 1 | 5 | 2 | 5 | 3 | -7 | +------+
striple(a, b, c)
6
vtriple(a, b, c)
1 +--------+ 1 | -267 | 2 | 204 | 3 | -3 | +--------+
end</lang>
Swift
<lang swift>import Foundation
infix operator • : MultiplicationPrecedence infix operator × : MultiplicationPrecedence
public struct Vector {
public var x = 0.0 public var y = 0.0 public var z = 0.0
public init(x: Double, y: Double, z: Double) {
(self.x, self.y, self.z) = (x, y, z)
}
public static func • (lhs: Vector, rhs: Vector) -> Double {
return lhs.x * rhs.x + lhs.y * rhs.y + lhs.z * rhs.z
}
public static func × (lhs: Vector, rhs: Vector) -> Vector {
return Vector(
x: lhs.y * rhs.z - lhs.z * rhs.y,
y: lhs.z * rhs.x - lhs.x * rhs.z,
z: lhs.x * rhs.y - lhs.y * rhs.x
)
}
}
let a = Vector(x: 3, y: 4, z: 5) let b = Vector(x: 4, y: 3, z: 5) let c = Vector(x: -5, y: -12, z: -13)
print("a: \(a)") print("b: \(b)") print("c: \(c)") print() print("a • b = \(a • b)") print("a × b = \(a × b)") print("a • (b × c) = \(a • (b × c))") print("a × (b × c) = \(a × (b × c))")</lang>
- Output:
a: Vector(x: 3.0, y: 4.0, z: 5.0) b: Vector(x: 4.0, y: 3.0, z: 5.0) c: Vector(x: -5.0, y: -12.0, z: -13.0) a • b = 49.0 a × b = Vector(x: 5.0, y: 5.0, z: -7.0) a • (b × c) = 6.0 a × (b × c) = Vector(x: -267.0, y: 204.0, z: -3.0)
Tcl
<lang tcl>proc dot {A B} {
lassign $A a1 a2 a3
lassign $B b1 b2 b3
expr {$a1*$b1 + $a2*$b2 + $a3*$b3}
} proc cross {A B} {
lassign $A a1 a2 a3
lassign $B b1 b2 b3
list [expr {$a2*$b3 - $a3*$b2}] \
[expr {$a3*$b1 - $a1*$b3}] \ [expr {$a1*$b2 - $a2*$b1}] } proc scalarTriple {A B C} {
dot $A [cross $B $C]
} proc vectorTriple {A B C} {
cross $A [cross $B $C]
}</lang> Demonstrating: <lang tcl>set a {3 4 5} set b {4 3 5} set c {-5 -12 -13} puts "a • b = [dot $a $b]" puts "a x b = [cross $a $b]" puts "a • b x c = [scalarTriple $a $b $c]" puts "a x b x c = [vectorTriple $a $b $c]"</lang>
Output:
a • b = 49 a x b = 5 5 -7 a • b x c = 6 a x b x c = -267 204 -3
uBasic/4tH
Since uBasic/4tH has only one single array, we use its variables to hold the offsets of the vectors. A similar problem arises when local vectors are required. <lang>a = 0 ' use variables for vector addresses b = a + 3 c = b + 3 d = c + 3
Proc _Vector (a, 3, 4, 5) ' initialize the vectors Proc _Vector (b, 4, 3, 5) Proc _Vector (c, -5, -12, -13)
Print "a . b = "; FUNC(_FNdot(a, b)) Proc _Cross (a, b, d) Print "a x b = (";@(d+0);", ";@(d+1);", ";@(d+2);")" Print "a . (b x c) = "; FUNC(_FNscalarTriple(a, b, c)) Proc _VectorTriple (a, b, c, d) Print "a x (b x c) = (";@(d+0);", ";@(d+1);", ";@(d+2);")" End
_FNdot Param (2) Return ((@(a@+0)*@(b@+0))+(@(a@+1)*@(b@+1))+(@(a@+2)*@(b@+2)))
_Vector Param (4) ' initialize a vector
@(a@ + 0) = b@ @(a@ + 1) = c@ @(a@ + 2) = d@
Return
_Cross Param (3)
@(c@+0) = @(a@ + 1) * @(b@ + 2) - @(a@ + 2) * @(b@ + 1) @(c@+1) = @(a@ + 2) * @(b@ + 0) - @(a@ + 0) * @(b@ + 2) @(c@+2) = @(a@ + 0) * @(b@ + 1) - @(a@ + 1) * @(b@ + 0)
Return
_FNscalarTriple Param (3)
Local (1) ' a "local" vector d@ = d + 3 ' (best effort) ;-) Proc _Cross(b@, c@, d@)
Return (FUNC(_FNdot(a@, d@)))
_VectorTriple Param(4)
Local (1) ' a "local" vector e@ = d + 3 ' (best effort) ;-) Proc _Cross (b@, c@, e@) Proc _Cross (a@, e@, d@)
Return</lang>
- Output:
a . b = 49 a x b = (5, 5, -7) a . (b x c) = 6 a x (b x c) = (-267, 204, -3) 0 OK, 0:1370
VBA
<lang vb>Option Base 1
Function dot_product(a As Variant, b As Variant) As Variant
dot_product = WorksheetFunction.SumProduct(a, b)
End Function
Function cross_product(a As Variant, b As Variant) As Variant
cross_product = Array(a(2) * b(3) - a(3) * b(2), a(3) * b(1) - a(1) * b(3), a(1) * b(2) - a(2) * b(1))
End Function
Function scalar_triple_product(a As Variant, b As Variant, c As Variant) As Variant
scalar_triple_product = dot_product(a, cross_product(b, c))
End Function
Function vector_triple_product(a As Variant, b As Variant, c As Variant) As Variant
vector_triple_product = cross_product(a, cross_product(b, c))
End Function
Public Sub main()
a = [{3, 4, 5}]
b = [{4, 3, 5}]
c = [{-5, -12, -13}]
Debug.Print " a . b = "; dot_product(a, b)
Debug.Print " a x b = "; "("; Join(cross_product(a, b), ", "); ")"
Debug.Print "a . (b x c) = "; scalar_triple_product(a, b, c)
Debug.Print "a x (b x c) = "; "("; Join(vector_triple_product(a, b, c), ", "); ")"
End Sub</lang>
- Output:
a . b = 49
a x b = (5, 5, -7)
a . (b x c) = 6
a x (b x c) = (-267, 204, -3)Visual Basic .NET
Class: Vector3D <lang vbnet>Public Class Vector3D
Private _x, _y, _z As Double
Public Sub New(ByVal X As Double, ByVal Y As Double, ByVal Z As Double)
_x = X
_y = Y
_z = Z
End Sub
Public Property X() As Double
Get
Return _x
End Get
Set(ByVal value As Double)
_x = value
End Set
End Property
Public Property Y() As Double
Get
Return _y
End Get
Set(ByVal value As Double)
_y = value
End Set
End Property
Public Property Z() As Double
Get
Return _z
End Get
Set(ByVal value As Double)
_z = value
End Set
End Property
Public Function Dot(ByVal v2 As Vector3D) As Double
Return (X * v2.X) + (Y * v2.Y) + (Z * v2.Z)
End Function
Public Function Cross(ByVal v2 As Vector3D) As Vector3D
Return New Vector3D((Y * v2.Z) - (Z * v2.Y), _
(Z * v2.X) - (X * v2.Z), _
(X * v2.Y) - (Y * v2.X))
End Function
Public Function ScalarTriple(ByVal v2 As Vector3D, ByVal v3 As Vector3D) As Double
Return Me.Dot(v2.Cross(v3))
End Function
Public Function VectorTriple(ByRef v2 As Vector3D, ByVal v3 As Vector3D) As Vector3D
Return Me.Cross(v2.Cross(v3))
End Function
Public Overrides Function ToString() As String
Return String.Format("({0}, {1}, {2})", _x, _y, _z)
End Function
End Class</lang> Module: Module1 <lang vbnet>Module Module1
Sub Main()
Dim v1 As New Vector3D(3, 4, 5)
Dim v2 As New Vector3D(4, 3, 5)
Dim v3 As New Vector3D(-5, -12, -13)
Console.WriteLine("v1: {0}", v1.ToString())
Console.WriteLine("v2: {0}", v2.ToString())
Console.WriteLine("v3: {0}", v3.ToString())
Console.WriteLine()
Console.WriteLine("v1 . v2 = {0}", v1.Dot(v2))
Console.WriteLine("v1 x v2 = {0}", v1.Cross(v2).ToString())
Console.WriteLine("v1 . (v2 x v3) = {0}", v1.ScalarTriple(v2, v3))
Console.WriteLine("v1 x (v2 x v3) = {0}", v1.VectorTriple(v2, v3))
End Sub
End Module</lang> Output:
v1: (3, 4, 5) v2: (4, 3, 5) v3: (-5, -12, -13) v1 . v2 = 49 v1 x v2 = (5, 5, -7) v1 . (v2 x v3) = 6 v1 x (v2 x v3) = (-267, 204, -3)
Wortel
<lang wortel>@let {
dot &[a b] @sum @mapm ^* [a b] cross &[a b] [[ -*a.1 b.2 *a.2 b.1 -*a.2 b.0 *a.0 b.2 -*a.0 b.1 *a.1 b.0 ]] scalarTripleProduct &[a b c] !!dot a !!cross b c vectorTripleProduct &[a b c] !!cross a !!cross b c a [3 4 5] b [4 3 5] c [5N 12N 13N] [[ !!dot a b !!cross a b @!scalarTripleProduct [a b c] @!vectorTripleProduct [a b c] ]]
}</lang> Returns:
[49 [5 5 -7] 6 [-267 204 -3]]
XPL0
<lang XPL0>include c:\cxpl\codes; \intrinsic 'code' declarations
func DotProd(A, B); \Return the dot product of two 3D vectors int A, B; \A ù B return A(0)*B(0) + A(1)*B(1) + A(2)*B(2);
proc CrossProd(A, B, C); \Calculate the cross product of two 3D vectors int A, B, C; \C:= A x B [C(0):= A(1)*B(2) - A(2)*B(1);
C(1):= A(2)*B(0) - A(0)*B(2); C(2):= A(0)*B(1) - A(1)*B(0);
]; \CrossProd
func ScalarTriProd(A, B, C); \Return the scalar triple product int A, B, C; \A ù (B x C) int D(3); [CrossProd(B, C, D); return DotProd(A, D); ]; \ScalarTriProd
proc VectTriProd(A, B, C, D); \Calculate the vector triple product int A, B, C, D; \D:= A x (B x C) int E(3); [CrossProd(B, C, E);
CrossProd(A, E, D);
]; \CrossProd
int A, B, C, D(3);
[A:= [3, 4, 5]; B:= [4, 3, 5]; C:= [-5, -12, -13];
IntOut(0, DotProd(A,B)); CrLf(0);
CrossProd(A, B, D); IntOut(0, D(0)); ChOut(0, 9\tab\); IntOut(0, D(1)); ChOut(0, 9\tab\); IntOut(0, D(2)); CrLf(0);
IntOut(0, ScalarTriProd(A,B,C)); CrLf(0);
VectTriProd(A, B, C, D); IntOut(0, D(0)); ChOut(0, 9\tab\); IntOut(0, D(1)); ChOut(0, 9\tab\); IntOut(0, D(2)); CrLf(0); ]</lang>
Output:
49 5 5 -7 6 -267 204 -3
zkl
Since the input vectors are all int, the output is int. For a float output, use float data (or convert) in the input vectors and change sum() to sum(0.0) (in dotp).
The [(a1,a2,a3)] parameter notation just means add a preamble to the function body to do list assignment: a1,a2,a3:=arglist[0]. Since we don't need the vector as such, don't bother to name it (in the parameter list) <lang zkl>fcn dotp(a,b){ a.zipWith('*,b).sum() } //1 slow but concise fcn crossp([(a1,a2,a3)],[(b1,b2,b3)]) //2
{ return(a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1) }</lang>
<lang zkl>a,b,c := T(3,4,5), T(4,3,5), T(-5,-12,-13); dotp(a,b).println(); //5 --> 49 crossp(a,b).println(); //6 --> (5,5,-7) dotp(a, crossp(b,c)).println(); //7 --> 6 crossp(a, crossp(b,c)).println(); //8 --> (-267,204,-3)</lang>
- Output:
49 L(5,5,-7) 6 L(-267,204,-3)
Or, using the GNU Scientific Library: <lang zkl>var [const] GSL=Import("zklGSL"); // libGSL (GNU Scientific Library) a:=GSL.VectorFromData( 3, 4, 5); b:=GSL.VectorFromData( 4, 3, 5); c:=GSL.VectorFromData(-5,-12,-13);
(a*b).println(); // 49, dot product a.copy().crossProduct(b) // (5,5,-7) cross product, in place
.format().println();
(a*(b.copy().crossProduct(c))).println(); // 6 scalar triple product (a.crossProduct(b.crossProduct(c))) // (-267,204,-3) vector triple product, in place
.format().println();</lang>
- Output:
49 5.00,5.00,-7.00 6 -267.00,204.00,-3.00
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