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


 
 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
 
 