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Frame A and point 

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body p

lie on the plane

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body z=0.3

:

Note that unit vector 

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body --uriencoded--z_%7B\mathrm%7BA%7D%7D

is pointing out of the screen. The coordinates of point 

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body p

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 

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body --uriencoded--%7B%7D%5e%7B\mathrm%7BA%7D%7D\!R_%7B\mathrm%7BB%7D%7D

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 

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body --uriencoded--p_%7B\mathrm%7BA%7D%7D = %7B%7D%5e%7B\mathrm%7BA%7D%7D\!R_%7B\mathrm%7BB%7D%7D \ p_%7B\mathrm%7BB%7D%7D

to compute

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body --uriencoded--p_%7B\mathrm%7BB%7D%7D

. Recall that rotation matrices are orthogonal, which means that

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body --uriencoded--\left( %7B%7D%5e%7B\mathrm%7BA%7D%7D\!R_%7B\mathrm%7BB%7D%7D \right)%5e%7B-1%7D = \left( %7B%7D%5e%7B\mathrm%7BA%7D%7D\!R_%7B\mathrm%7BB%7D%7D \right)%5e%7B\operatorname%7BT%7D%7D

.

(c) Now suppose the origin of frame B is at 

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body --uriencoded--3.7 x_%7B\mathrm%7BA%7D%7D + 1.1 y_%7B\mathrm%7BA%7D%7D + 0 z_%7B\mathrm%7BA%7D%7D

relative to the origin of frame A. What is the transformation matrix 

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body --uriencoded--%7B%7D%5e%7B\mathrm%7BA%7D%7DT_%7B\mathrm%7BB%7D%7D

that relates frames A and B?

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

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body --uriencoded--p_%7B\mathrm%7BB%7D%7D

:

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alignment center
\begin{Bmatrix}p_{\mathrm{A}}\\1\end{Bmatrix} = {}^{\mathrm{A}}T_{\mathrm{B}} \begin{Bmatrix}p_{\mathrm{B}}\\1\end{Bmatrix}

(e) Sketch frame A, frame B, and point 

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body p

to confirm that your answer to part (d) is correct.

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Include Page S7.3 Introduction to transformation matrices S7.3 Introduction to transformation matrices