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

A p p l y t h e r o t a t i o n o f c o o r d i n a t e s g i v e n

b y t h e f o l l o w i n g m a t r i x t o t h e e q u a t i o n

x y = 1; w h a t i s t h e e q u a t i o n i n t h u v c o o r d i n a t e

s y s t e m ?

[\begin{matrix} u \\ v \end{matrix}] = [\begin{matrix} c o s 45 s i n 45 \\ - s i n 45 c o s 45 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}]

Answered by anurup last updated on 16/Feb/23

[\begin{matrix} x \\ y \end{matrix}] = {[\begin{matrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{matrix}]}^{- 1} [\begin{matrix} u \\ v \end{matrix}]

= [\begin{matrix} \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{matrix}] [\begin{matrix} u \\ v \end{matrix}]

∴ x = \frac{1}{\sqrt{2}} (u - v)

y = \frac{1}{\sqrt{2}} (u + v)

x y = 1 \Rightarrow \frac{1}{2} (u^{2} - v^{2}) = 1 \Rightarrow u^{2} - v^{2} = 2

Commented by Humble last updated on 16/Feb/23

T h a n k y o u, s i r

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