91Ó°ÊÓ

A simple form for a rotation matrix around the y-axis is$\mathbf{}\mathit{A}\mathbf{=}\mathbf{\left(}\begin{array}{ccc}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{θ}& \mathbf{0}& \mathbf{s}\mathbf{i}\mathbf{n}\mathbf{θ}\\ \mathbf{0}& \mathbf{1}& \mathbf{0}\\ \mathbf{-}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{θ}& \mathbf{0}& \mathbf{c}\mathbf{o}\mathbf{s}\mathbf{θ}\end{array}\mathbf{\right)}$ .

The matrix corresponding to a rotation of ${90}^{∘}$ about the y-axis together with a reflection through the $\left(x,z\right)$ plane is to be determined.

Take a general rotation matrix by angle $\mathrm{θ}$ around the y-axis.

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

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

role="math" localid="1658995486018" $\left(\begin{array}{ccc}\mathrm{³¦´Ç²õθ}& 0& \mathrm{²õ¾\pm ²Ôθ}\\ 0& 1& 0\\ -\mathrm{²õ¾\pm ²Ôθ}& 0& \mathrm{³¦´Ç²õθ}\end{array}\right)\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)=\left(\begin{array}{c}\mathrm{³¦´Ç²õθ}\\ 0\\ -\mathrm{²õ¾\pm ²Ôθ}\end{array}\right)$

This result is expected.

Combine this with reflection, which gives $\left(\begin{array}{ccc}\mathrm{³¦´Ç²õθ}& 0& -\mathrm{²õ¾\pm ²Ôθ}\\ 0& -1& 0\\ \mathrm{²õ¾\pm ²Ôθ}& 0& \mathrm{³¦´Ç²õθ}\end{array}\right)$.

Substitute $\mathrm{θ}={90}^{∘}$ in the matrixrole="math" localid="1658995357046" $\left(\begin{array}{ccc}\mathrm{³¦´Ç²õθ}& 0& -\mathrm{²õ¾\pm ²Ôθ}\\ 0& -1& 0\\ \mathrm{²õ¾\pm ²Ôθ}& 0& \mathrm{³¦´Ç²õθ}\end{array}\right)$

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

The matrix corresponding to a rotation of ${90}^{∘}$ about theyaxis together with a reflection through the$\left(x,z\right)$ plane is$\left(\begin{array}{ccc}0& 0& 1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right)$ .