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Point P(x, y) is rotated by an angle the...

Point P(x, y) is rotated by an angle `theta` in anticlockwise direction. The new position of point P is `Q (x_(1), y_(1))`. If `[(x_(1)),(y_(1))]=A[(x),(y)]`, then find matrix A.

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The correct Answer is:
`A=[(cos theta,-sin theta),(sin theta,cos theta)]`

From the figure, `x_(1)=r cos (theta+alpha)`
`=(r cos alpha) cos theta-(r sin alpha) sin theta`
`:. X_(1)=x cos theta -y sin theta`
and `y_(1)= r sin (theta +alpha)=(r sin alpha) cos theta+(r cos alpha) sin theta`
`:. Y_(1)=x sin theta +y cos theta`
So, `[(x_(1)),(y_(1))]=[(cos theta, -sin theta),(sin theta, cos theta)][(x),(y)]`
`:. A=[(cos theta, -sin theta),(sin theta, cos theta)]`
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