Find the coordinates of M in the original sys | vedclass

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(B) Let the original coordinates be $(x, y)$ and the new coordinates be $(x', y') = (4, -3)$. The rotation angle is $	heta = 135^{\circ}$.The transfo

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$\left(\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)$

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(B) Let the original coordinates be $(x, y)$ and the new coordinates be $(x', y') = (4, -3)$. The rotation angle is $	heta = 135^{\circ}$.The transformation formulas for rotation of axes are:$x = x' \cos 	heta - y' \sin 	heta$$y = x' \sin 	heta + y' \cos 	heta$Given $\cos 135^{\circ} = -\frac{1}{\sqrt{2}}$ and $\sin 135^{\circ} = \frac{1}{\sqrt{2}}$,we substitute these values:$x = 4 \left(-\frac{1}{\sqrt{2}}ight) - (-3) \left(\frac{1}{\sqrt{2}}ight)$$x = -\frac{4}{\sqrt{2}} + \frac{3}{\sqrt{2}} = -\frac{1}{\sqrt{2}}$$y = 4 \left(\frac{1}{\sqrt{2}}ight) + (-3) \left(-\frac{1}{\sqrt{2}}ight)$$y = \frac{4}{\sqrt{2}} + \frac{3}{\sqrt{2}} = \frac{7}{\sqrt{2}}$Thus,the original coordinates are $\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}ight)$.

Find the coordinates of $M$ in the original system if the point $M$ changes to $(4, -3)$ when the axes are rotated through an angle of $135^{\circ}$.

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