7.3 Introduction to transformation matrices

Frame A and point  lie on the plane :

Note that unit vector  is pointing out of the screen. The coordinates of point  are (2.1, 0.5, 0) when expressed in frame A. Frame B lies in the same plane and has the orientation shown below:

(a) Find the rotation matrix  that relates the orientation of frames A and B. Use any strategy to check your answer and explain why it is correct.

(b) For part (b) only, suppose the origins of frames A and B are coincident. Use your answer from part (a) and the expression  to compute . Recall that rotation matrices are orthogonal, which means that .

(c) Now suppose the origin of frame B is at  relative to the origin of frame A. What is the transformation matrix  that relates frames A and B?

(d) Use your answer from part (c) and the expression below to compute :

(e) Sketch frame A, frame B, and point  to confirm that your answer to part (d) is correct.