PLS EXPLAIN ME ROTATION OF MATRIX THROUGH ANGLE 0(theta) ...also give an example ..!!!

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{To} \mathrm{rotate} \mathrm{a} \mathrm{column} \mathrm{matrix} \left(\mathrm{also} \mathrm{called} \mathrm{a} \mathrm{column} \mathrm{vector}\right), \mathrm{through} \mathrm{an} \mathrm{angle} \mathrm{\theta }, \mathrm{we} \mathrm{multyply} \mathrm{it} \mathrm{by} \mathrm{the} \mathrm{rotation} \mathrm{matrix} \mathrm{R}\left(\mathrm{\theta }\right). \\ \mathrm{In} \mathrm{two} \mathrm{dimensions}, \mathrm{the} \mathrm{rotation} \mathrm{matrix} \mathrm{for} \mathrm{angle} \mathrm{\theta } \mathrm{has} \mathrm{the} \mathrm{following} \mathrm{form}: \\ \mathrm{R}\left(\mathrm{\theta }\right)=\left[\begin{array}{cc}\mathrm{cos\theta }& -\mathrm{sin\theta }\\ \mathrm{sin\theta }& \mathrm{cos\theta }\end{array}\right] \\ \mathrm{Suppose} \mathrm{we} \mathrm{have} \mathrm{a} \mathrm{vector} \stackrel{\rightharpoonup }{\mathrm{v}}=2\hat{\mathrm{i}}+3\hat{\mathrm{j}} \\ \mathrm{Or}, \stackrel{\rightharpoonup }{\mathrm{v}}=\left(\begin{array}{c}{\mathrm{v}}_{\mathrm{x}}\\ {\mathrm{v}}_{\mathrm{y}}\end{array}\right)=\left(\begin{array}{c}2\\ 3\end{array}\right) \\ \mathrm{If} \mathrm{we} \mathrm{want} \mathrm{to} \mathrm{rotate} \mathrm{this} \mathrm{vector} \mathrm{by}, \mathrm{say}, 30°, \mathrm{we} \mathrm{multiply} \mathrm{the} \mathrm{vector} \mathrm{by} \mathrm{R}\left(30°\right). \\ \mathrm{That} \mathrm{is}, \mathrm{the} \mathrm{rotated} \mathrm{vector}, \\ \stackrel{\rightharpoonup }{\mathrm{v}\text{'}}=\mathrm{R}\left(\mathrm{\theta }\right)\times \stackrel{\rightharpoonup }{\mathrm{v}} \\ \mathrm{Or}, \\ \left(\begin{array}{c}{\mathrm{v}}_{\mathrm{x}}\text{'}\\ {\mathrm{v}}_{\mathrm{y}}\text{'}\end{array}\right)=\left[\begin{array}{cc}\mathrm{cos\theta }& -\mathrm{sin\theta }\\ \mathrm{sin\theta }& \mathrm{cos\theta }\end{array}\right]\times \left(\begin{array}{c}{\mathrm{v}}_{\mathrm{x}}\\ {\mathrm{v}}_{\mathrm{y}}\end{array}\right) \\ \left(\begin{array}{c}{\mathrm{v}}_{\mathrm{x}}\text{'}\\ {\mathrm{v}}_{\mathrm{y}}\text{'}\end{array}\right)=\left[\begin{array}{cc}\cos \left(30°\right)& -\sin \left(30°\right)\\ \sin \left(30°\right)& \cos \left(30°\right)\end{array}\right]\times \left(\begin{array}{c}2\\ 3\end{array}\right) \\ \left(\begin{array}{c}{\mathrm{v}}_{\mathrm{x}}\text{'}\\ {\mathrm{v}}_{\mathrm{y}}\text{'}\end{array}\right)=\left[\begin{array}{cc}\left(\frac{\sqrt{3}}{2}\right)& \left(-\frac{1}{2}\right)\\ \left(\frac{1}{2}\right)& \left(\frac{\sqrt{3}}{2}\right)\end{array}\right]\times \left(\begin{array}{c}2\\ 3\end{array}\right) \\ \left(\begin{array}{c}{\mathrm{v}}_{\mathrm{x}}\text{'}\\ {\mathrm{v}}_{\mathrm{y}}\text{'}\end{array}\right)=\left(\begin{array}{c}\left(\frac{\sqrt{3}}{2}\right)\times 2+\left(-\frac{1}{2}\right)\times 3\\ \left(\frac{1}{2}\right)\times 2+\left(\frac{\sqrt{3}}{2}\right)\times 3\end{array}\right) \\ \left(\begin{array}{c}{\mathrm{v}}_{\mathrm{x}}\text{'}\\ {\mathrm{v}}_{\mathrm{y}}\text{'}\end{array}\right)=\left(\begin{array}{c}\left(\sqrt{3}-\frac{3}{2}\right)\\ \left(1+\frac{3\sqrt{3}}{2}\right)\end{array}\right) \\ \left(\begin{array}{c}{\mathrm{v}}_{\mathrm{x}}\text{'}\\ {\mathrm{v}}_{\mathrm{y}}\text{'}\end{array}\right)\approx \left(\begin{array}{c}0.23205\\ 3.5981\end{array}\right) \\ \mathrm{The} \mathrm{rotated} \mathrm{vector} \mathrm{is}, \\ \stackrel{\rightharpoonup }{\mathrm{v}\text{'}}=0.23205\hat{\mathrm{i}}+3.5981\hat{\mathrm{j}} \\ \mathrm{Refer} \mathrm{the} \mathrm{figure}. \mathrm{The} \mathrm{vectors} \mathrm{before} \mathrm{and} \mathrm{after} \mathrm{rotation} \mathrm{are} \mathrm{shown} \mathrm{in} \mathrm{it}. \end{gathered}$$

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