Question

4. Prove that the rotation matrix about y-axis and z-axis can be presented as follows (20 pt) $\begin{aligned} R_{y, \theta} &= \begin{bmatrix} \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ -\sin \theta & 0 & \cos \theta \end{bmatrix} \\ R_{z, \theta} &= \begin{bmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{aligned}$

          4. Prove that the rotation matrix about y-axis and z-axis can be presented as follows (20 pt)
$\begin{aligned}
R_{y, \theta} &= \begin{bmatrix}
\cos \theta & 0 & \sin \theta \\
0 & 1 & 0 \\
-\sin \theta & 0 & \cos \theta
\end{bmatrix} \\
R_{z, \theta} &= \begin{bmatrix}
\cos \theta & -\sin \theta & 0 \\
\sin \theta & \cos \theta & 0 \\
0 & 0 & 1
\end{bmatrix}
\end{aligned}$

4. Prove that the rotation matrix about y-axis and z-axis can be presented as follows (20 pt)
Ry, θ =
< b m a t r i x >

Rz, θ =
< b m a t r i x >

Added by Peggy B.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Thane Stiles

06/10/2020

Step 1

A rotation matrix is a matrix that rotates a vector in a 3D space. The rotation matrix about y-axis rotates a vector around the y-axis, and the rotation matrix about z-axis rotates a vector around the z-axis. Show more…

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4. Prove that the rotation matrix about y-axis and z-axis can be presented as follows (20 pt) $$R_{y,\theta} = \begin{bmatrix} cos\theta & 0 & sin\theta \\ 0 & 1 & 0 \\ -sin\theta & 0 & cos\theta \end{bmatrix}$$ $$R_{z,\theta} = \begin{bmatrix} cos\theta & -sin\theta & 0 \\ sin\theta & cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$$