91Ó°ÊÓ

A simple form for a rotation matrix around the x-axis is $\mathbf{A}\mathbf{=}\left(\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{³¦´Ç²õθ}& -\mathrm{²õ¾\pm ²Ôθ}\\ 0& \mathrm{²õ¾\pm ²Ôθ}& \mathrm{³¦´Ç²õθ}\end{array}\right)$ .

The matrices which produce a rotation $\mathrm{θ}$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 $\mathrm{θ}$around the x-axis

$A=\left(\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{³¦´Ç²õθ}& -\mathrm{²õ¾\pm ²Ôθ}\\ 0& \mathrm{²õ¾\pm ²Ôθ}& \mathrm{³¦´Ç²õθ}\end{array}\right)$

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

$\left(\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{³¦´Ç²õθ}& -\mathrm{²õ¾\pm ²Ôθ}\\ 0& \mathrm{²õ¾\pm ²Ôθ}& \mathrm{³¦´Ç²õθ}\end{array}\right)\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)$

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

$\left(\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{³¦´Ç²õθ}& -\mathrm{²õ¾\pm ²Ôθ}\\ 0& \mathrm{²õ¾\pm ²Ôθ}& \mathrm{³¦´Ç²õθ}\end{array}\right)\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)=\left(\begin{array}{c}1\\ \cos \mathrm{θ}\\ \sin \mathrm{θ}\end{array}\right)$

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.

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