#|
				Quiz 3
|#

;; #1
(message "
Finding a norm of a difference:
")
(def v (vector 4 -7))
(def w (vector -2 9))
(norm (calc v - w))

;; #2
(message "
Ahoy Mates! Cap'n Flint wants to know: where did I put that treasure?
")
(def v (vector 0 30))
(def w (vector 25 25)) ;; 45 degree angle!
(def x (vector 0 -10))
(norm (calc v + w + x))

;; #3
(message "
Planes, and vectors perpendicular and parallel to them:
")
(message "
Strategy:

	Rewrite the equation of the plane as

		3*(x - 0) + -1*(y - 0) + -1*(z - 2) = 0

Thus (0,0,2) is a point in the plane, and form a vector of any point in the
plane, e.g. (x,y,z), and (0,0,2), which is perpendicular to (3,-1,-1) and hence
parallel to the plane. So, for example, setting z=2 and x=1 gives y=3, so
(1,3,0) is a vector parallel to the plane.
")
(def perp (vector 3 -1 -1))
(def para (vector 1 3 0))
(dot para perp)

(message "
Here's another way of getting at it: by finding the zero of a function:
")
(def x 1)
(def y 1)
(func f(z) 3 * (x - 0) + -1 * (y - 0) + -1 * (z - 2))
(roots f 0)

(message "
I tested your parallel vectors using this function:
")
(defun vec-tester (x y z)
  (print (vector x y z))
  (dot perp (vector x y z))
  )
(alias3 vt vec-tester)
(vt 1 3 3)
(vt 1 0 3)
(vt 2 3 3)
(vt 1 3 -1)
(vt 1 2 1)
(vt 4 4 2)


(message "
Some of you looked for other points in the plane: I tested those with a plane
function:
")
(func plane(x y z) 3 * (x - 0) + -1 * (y - 0) + -1 * (z - 2))
;; 
(plane 1 -1 6)
(plane -1 -3 0)
;; 
(plane 1 1 5)
(plane 2 4 4)
;; 
(plane 1 2 3)