SICP Exercise 2.46: Vectors

A two-dimensional vector v running from the origin to a point can be represented as a pair consisting of an x-coordinate and a y-coordinate. Implement a data abstraction for vectors by giving a constructor make-vect and corresponding selectors xcor-vect and ycor-vect. In terms of your selectors and constructor, implement procedures add-vect, sub-vect, and scale-vect that perform the operations vector addition, vector subtraction, and multiplying a vector by a scalar:

(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)
(x1, y1) - (x2, y2) = (x1 - x2, y1 - y2)
        s · (x, y) = (sx, sy)

As a two-dimensional vector has only two pieces of data associated with it, the x- and y-coordinates, it lends itself naturally to being represented as a pair. Let's put x as the first elements, and y as the second. We can then access them via car and cdr respectively:

(define (make-vect x y) (cons x y))
(define (xcor-vect v) (car v))
(define (ycor-vect v) (cdr v))

Given this representation (or at least this constructor and these selectors), we can implement the given operations straightforwardly. We simply need to extract the x- and y-coordinates from the vectors we're dealing with using the selectors, perform the appropriate mathematical operations on them and then create a new vector as our result using the constructor:

(define (add-vect v1 v2)
  (make-vect (+ (xcor-vect v1) (xcor-vect v2))
             (+ (ycor-vect v1) (ycor-vect v2))))

(define (sub-vect v1 v2)
  (make-vect (- (xcor-vect v1) (xcor-vect v2))
             (- (ycor-vect v1) (ycor-vect v2))))

(define (scale-vect s v)
  (make-vect (* s (xcor-vect v))
             (* s (ycor-vect v))))

Let's see them in action:

> (define v1 (make-vect 4 5))
> (define v2 (make-vect 7 3))

> v1
'(4 . 5)
> v2
'(7 . 3)

> (xcor-vect v1)
4
> (ycor-vect v1)
5
> (ycor-vect v2)
3
> (xcor-vect v2)
7

> (add-vect v1 v2)
'(11 . 8)
> (add-vect v2 v1)
'(11 . 8)

> (sub-vect v1 v2)
'(-3 . 2)
> (sub-vect v2 v1)
'(3 . -2)

> (scale-vect 3 v1)
'(12 . 15)
> (scale-vect 2 v2)
'(14 . 6)