Code

(define-struct quaternion (cc ic jc kc))
;; A Quaternion is a (make-quaternion Num Num Num Num)

;; Purpose:
;; (quat-mult a b q1 q2) consumes two numbers a, b and two Quaternions q1, q2
;; and produces the result of the multiplication q1 and q2

;; Contracts:
;; quat-mult: Num Num Quaternion Quaternion -> Quaternion
;; requires: a, b both not equal to 0.

;; Examples:
(check-expect (quat-mult 0 5 (make-quaternion 1 2 3 4)
                    (make-quaternion 5 6 7 8)) (make-quaternion 110 36 22 24))
(check-expect (quat-mult 1 -1 (make-quaternion 2 3 4 8)
                    (make-quaternion 2 8 9 0)) (make-quaternion -8 -50 -38 11))


(define (quat-mult a b q1 q2)
  
  (make-quaternion
    (- (+
     (* (quaternion-cc q1) (quaternion-cc q2))
     (* (quaternion-jc q1) (quaternion-jc q2) b)    
     (* (quaternion-ic q1) (quaternion-ic q2) a))
     (* (quaternion-kc q1) (quaternion-kc q2)  a b))
    (- (+
        (* (quaternion-cc q1) (quaternion-ic q2))
        (* (quaternion-ic q1) (quaternion-cc q2))   
        (* (quaternion-kc q1) (quaternion-jc q2) b))
       (* (quaternion-jc q1) (quaternion-kc q2) b))
    (- (+
        (* (quaternion-cc q1) (quaternion-jc q2))
        (* (quaternion-ic q1) (quaternion-kc q2) a)
        (* (quaternion-jc q1) (quaternion-cc q2)))
       (* (quaternion-kc q1) (quaternion-ic q2) a))
    (- (+
        (* (quaternion-cc q1) (quaternion-kc q2))
        (* (quaternion-ic q1) (quaternion-jc q2))
        (* (quaternion-kc q1) (quaternion-cc q2)))
       (* (quaternion-jc q1) (quaternion-ic q2)))))


;; Tests:
(check-expect (quat-mult 1 5 (make-quaternion 2 0 0 0)
                         (make-quaternion 2 8 9 0)) (make-quaternion 4 16 18 0))
(check-expect (quat-mult 1 0 (make-quaternion 2 4 3 8)
                         (make-quaternion 2 0 2 0)) (make-quaternion 4 8 10 24))
(check-expect (quat-mult 2 5 (make-quaternion 2 3 9 1)
                    (make-quaternion 5 6 7 2)) (make-quaternion 341 -28 59 -24))