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


(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)