Question

Write the matrices which produce a rotation about the axis, or that rotation combined with a reflection through the (y,z) plane.

Short answer

The matrix of rotation around the x-axis is $\left(\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos\theta }& -\mathrm{sin\theta }\\ 0& \mathrm{sin\theta }& \mathrm{cos\theta }\end{array}\right)$and the matrix of rotation around the x axis combined with the reflection through the$\left(y,z\right)$ plane is$\left(\begin{array}{ccc}-1& 0& 0\\ 0& \mathrm{cos\theta }& -\mathrm{sin\theta }\\ 0& \mathrm{sin\theta }& \mathrm{cos\theta }\end{array}\right)$ .

Step-by-step solution

Rotation matrix

A simple form for a rotation matrix around the x-axis is $\mathbf{A}\mathbf{=}\left(\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos\theta }& -\mathrm{sin\theta }\\ 0& \mathrm{sin\theta }& \mathrm{cos\theta }\end{array}\right)$ .

Find the matrix of rotation

The matrices which produce a rotation $\theta$about the x-axis, or that rotation combined with a reflection through the (y,z) plane are to be determined.

Take a general rotation matrix by angle $\theta$around the x-axis

$A=\left(\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos\theta }& -\mathrm{sin\theta }\\ 0& \mathrm{sin\theta }& \mathrm{cos\theta }\end{array}\right)$

Verify the general rotation matrix by action on the vector i .

$\left(\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos\theta }& -\mathrm{sin\theta }\\ 0& \mathrm{sin\theta }& \mathrm{cos\theta }\end{array}\right)\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)$

Again, verify A by acting on a vector in (y,z) the plane.

$\left(\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos\theta }& -\mathrm{sin\theta }\\ 0& \mathrm{sin\theta }& \mathrm{cos\theta }\end{array}\right)\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)=\left(\begin{array}{c}1\\ \cos \theta \\ \sin \theta \end{array}\right)$

The matric of reflection through the yz-plane is given as $\left(\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$.

Verify it by taking any vector $\left(x,y,z\right)$and acting on it.

$\left(\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}-x\\ y\\ z\end{array}\right)$

Now, evaluate the combined matrix of rotation around the x axis and reflection through the yz-plane.

$\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \theta & -\sin \theta \\ 0& \sin \theta & \cos \theta \end{array}\right)\left(\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}-1& 0& 0\\ 0& \cos \theta & -\sin \theta \\ 0& \sin \theta & \cos \theta \end{array}\right)$