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


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 ]