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

   Dot product
   Quaternion type


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);
]