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 = 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