Question

Construct the matrix corresponding to a rotation of ${\mathbf{90}}^{\mathbf{\circ }}$about the yaxis together with a reflection through the $\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{z}\mathbf{)}$plane.

Short answer

The matrix corresponding to a rotation of ${90}^{\circ }$ about theyaxis together with a reflection through the$\left(x,y\right)$ plane is role="math" localid="1658995152787" $\left(\begin{array}{ccc}0& 0& 1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right)$ .

Step-by-step solution

Rotation matrix

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

 Find the matrix of rotation

The matrix corresponding to a rotation of ${90}^{\circ }$ 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 $\theta$ around the y-axis.

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

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

role="math" localid="1658995486018" $\left(\begin{array}{ccc}\mathrm{cos\theta }& 0& \mathrm{sin\theta }\\ 0& 1& 0\\ -\mathrm{sin\theta }& 0& \mathrm{cos\theta }\end{array}\right)\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)=\left(\begin{array}{c}\mathrm{cos\theta }\\ 0\\ -\mathrm{sin\theta }\end{array}\right)$

This result is expected.

Combine this with reflection, which gives $\left(\begin{array}{ccc}\mathrm{cos\theta }& 0& -\mathrm{sin\theta }\\ 0& -1& 0\\ \mathrm{sin\theta }& 0& \mathrm{cos\theta }\end{array}\right)$.

Substitute $\theta ={90}^{\circ }$ in the matrixrole="math" localid="1658995357046" $\left(\begin{array}{ccc}\mathrm{cos\theta }& 0& -\mathrm{sin\theta }\\ 0& -1& 0\\ \mathrm{sin\theta }& 0& \mathrm{cos\theta }\end{array}\right)$

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

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