4. Frame {B} is obtained by rotating frame {A} about axis Z by 30. Given Ap = [3] 5, find Bp. (5 points) 5. Reference frames {A} and {B} are shown below. 1) Find the rotation matrix AR by identifying its three column vectors; (5 points) 2) Find the rotation matrix BR by identifying its three column vectors; (5 points) and 3) Calculate BRT and BR-1. (5 points) ZA YB ZB YA 6. Reference frame {B} is obtained from reference frame {A} by performing the following three rotations: rotate about Z axis by angle y, rotate about the new X axis (named X' axis) by angle a, rotate about the new Y axis (named Y"axis) by angle . 1) Create three drawings to illustrate the three rotations (6 points). 2) Identify the rotation matrix corresponding to each rotation and the rotation matrix describing the rotation of {B} relative to {A} (9 points). 7. Frame {B} is obtained from frame {A} with a 123 sequence of Euler angles (i.e., rotate about X axis by angle a, rotate about the new Y axis by angle , rotate about the new Z axis by angle y). 1) Derive the rotation matrix R describing the rotation from {B} to {A} in terms of a, , and y. (9 points) 2) If the corresponding rotation matrix R from {B} to {A} is given below (i.e., all elements rij,i = 1,2,3 and j = 1,2,3 are known), derive the formulae to compute the three Euler angles (i.e., express the three Euler angles in terms of the elements ri). Indicate the conditions, if any, under which the formulae do not exist. (6 points) 11 112 113 R=21 21 22 123 31 32 33 8. Find the rotation matrix from {B} to {A}. {B} is obtained by conducting the following three rotations: 90 about the Z axis, then -90 about the Y' axis, lastly -90 about the X" axis. 1) Create three drawings to illustrate the three rotations (6 points). 2) Calculate the rotation matrix describing the rotation from {B} to {A} (9 points). 10. Find the rotation axis k and rotation angle 0 corresponding to the following rotation matrix. (10 points) 0.5 0.5 1/2 R(k, 0) = 0.5 0.5 -1/2 -1/2 1/2 0