Problem Statement

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: then the following common vector products are defined:

Given the three vectors:

Let's start with the solution:

(lib 'math)

(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))

(cross-product a b)
    → #( 5 5 -7)
(dot-product a b)
    → 49
(scalar-triple-product a b c)
    → 6
(vector-triple-product a b c)
    → #( -267 204 -3)