1) Prove the matrix below is representing a rotation of a frame about the Y-axis by B

1) Prove the matrix below is representing a rotation of a frame about the Y-axis by B. (10 points) COS 0 0 sin 0 Ry(0) =[ 0 1 0 sin e 0 cos 0 
2) Frame )[31 is located coincident with frame {Al. We rotate )[31 about ZB-axis by 30°, and then we rotate the resulting frame about Ye-axis by -30°. Find the resulting rotation matrix, 1?`,' that combines both rotations. (10 points) 
3) Prove Rx(a)Rx(13) = Rx(a-F13) using rotation matrices about principle axes and trigonometric identities in the handout given. (10 points) 
4) A velocity vector and a transformation matrix are given below. Find P, the velocity vector expressed in {Al. (10 points) 
[t0.0 V.= I 20.0 . 30.0 
[0.866 —0.500 0.000 11.0 il H = 0.000 00..086006 01..000000 —3.0 0 0 0 1 
5) For the given matrix below, find al. (20 points) 
[ 0.25 0.43 0.86 5.0 H° = 0.87 —0.50 0.00 —4.0 0.43 0.75 —0.50 3.0 0 0 0 1 
6) (Optional - Extra credit) For H in problem 5) and 11?=11,1 in problem 4), find H. (5 points)