Question

Show that the $\mathit{C}$ matrix in (11.10) does represent a rotation by finding the rotation angle. Write equations (7.13) and (11.13) for this rotation.

Short answer

$$\begin{gathered} x=x^{\text{'}}\cos 63.43-y^{\text{'}}\sin 63.43 \\ y=x^{\text{'}}\sin 63.43-y^{\text{'}}\cos 63.43 \end{gathered}$$

Step-by-step solution

Given information

Matrix$C=\left(\begin{array}{c}\frac{1}{\sqrt{5}}\\ \frac{2}{\sqrt{5}}\end{array}\begin{array}{c}\frac{-2}{\sqrt{5}}\\ \frac{1}{\sqrt{5}}\end{array}\right)$

Determinant of a matrix

The signed factor by which this matrix scales regions is called the determinant of a matrix. If the sign is negative, the matrix's orientation is reversed.

Calculate the determinant of  C

The determinant of $C$.

$$\begin{gathered} detC=\left|\begin{array}{cc}\frac{1}{\sqrt{5}}& \frac{-2}{\sqrt{5}}\\ \frac{2}{\sqrt{5}}& \frac{1}{\sqrt{5}}\end{array}\right| \\ =\frac{1}{5}-\left(-\frac{4}{5}\right) \\ =1 \end{gathered}$$

Matrix $C$ represents a rotation.

Equate matrix $C$ by the general rotation matrix.

$$\begin{gathered} \cos \theta =\frac{1}{\sqrt{5}} \\ \theta =63.43° \end{gathered}$$

Substitute the value of  θ

Substitute the value of $\theta$.

$\left(\begin{array}{c}X\\ Y\end{array}\right)=\left(\begin{array}{c}\cos 63.43\\ \sin 63.43\end{array}\begin{array}{c}-\sin 63.43\\ \cos 63.43\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$

This gives,

$$\begin{gathered} x=x^{\text{'}}\cos 63.43-y^{\text{'}}\sin 63.43 \\ y=x^{\text{'}}\sin 63.43-y^{\text{'}}\cos 63.43 \end{gathered}$$