Apply-the-rotation-of-coordinates-given-by-the-following-matrix-to-the-equation-xy-1-what-is-the-equation-in-th-uv-coordinate-system-u-v-cos45-sin45-sin45-co

Question Number 187335 by Humble last updated on 16/Feb/23

$${Apply}\:{the}\:{rotation}\:{of}\:{coordinates}\:{given} \\ $$$${by}\:{the}\:{following}\:{matrix}\:{to}\:{the}\:{equation}\: \\ $$$${xy}=\mathrm{1};\:{what}\:{is}\:{the}\:{equation}\:{in}\:{th}\:{uv}\:{coordinate}\: \\ $$$${system}? \\ $$$$\begin{bmatrix}{{u}}\\{{v}}\end{bmatrix}=\begin{bmatrix}{{cos}\mathrm{45}\:\:\:\:\:\:\:\:\:\:\:\:\:\:{sin}\mathrm{45}}\\{−{sin}\mathrm{45}\:\:\:\:\:\:\:\:\:\:{cos}\mathrm{45}}\end{bmatrix}\begin{bmatrix}{{x}}\\{{y}}\end{bmatrix} \\ $$

Answered by anurup last updated on 16/Feb/23

$$\begin{bmatrix}{{x}}\\{{y}}\end{bmatrix}=\:\begin{bmatrix}{\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}}&{\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}}\\{−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}}&{\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}}\end{bmatrix}^{−\mathrm{1}} \begin{bmatrix}{{u}}\\{{v}}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:=\begin{bmatrix}{\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}}&{−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}}\\{\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}}&{\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}}\end{bmatrix}\begin{bmatrix}{{u}}\\{{v}}\end{bmatrix} \\ $$$$\therefore{x}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}\:}\left({u}−{v}\right) \\ $$$${y}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\left({u}+{v}\right) \\ $$$${xy}=\mathrm{1}\Rightarrow\frac{\mathrm{1}}{\mathrm{2}}\left({u}^{\mathrm{2}} −{v}^{\mathrm{2}} \right)=\mathrm{1}\Rightarrow{u}^{\mathrm{2}} −{v}^{\mathrm{2}} =\mathrm{2} \\ $$$$ \\ $$

Commented by Humble last updated on 16/Feb/23

$${Thank}\:{you},\:{sir} \\ $$

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