Question 4: (20 pts)

For this question, you may use floating points calculations. For the following problems, you may assume that scaling is left unchanged (ie: S = I)

a) (5 pts) Give the transform W1 required to move an object so that its position is p1 and its forward vector f1 and up vector u1 are aligned with the values below.

p1 = (-5, -3, -2) f1 = (-3, 3, -4) u1 = (9, 25, 12)

b) (2 pts) Express W1 as a product of a rotation matrix R1 and a translation matrix T1 .

c) (3 pts) Give the transform W2 required to move an object so that its position p2 , forward vector f2 and up vector u2 are aligned with these values:

p2 = (1, -1, -4) f2 = (-4, 4, 1) u2 = (16, 17, -4)

d) (5 pts) An object has the position and orientation listed in a) and you need to move it to the position and orientation listed in c). Give the transform W12 that will acccomplish this.

Note: Be very clear about how you construct W12.

e) (2 pts) Show that W12 correctly transforms p1 into p2 .

f) (3 pts) Show that W12 correctly transforms f1 (and u1 ) into vectors having the same length as f1 (or u1 ) but in the direction of f2 (or u2 ).