Question

The point $(4,1)$ undergoes the following three transformations successively:
(i) Reflection about the line $y=x$
(ii) Transformation through a distance $2$ units along the positive direction of x-axis
(iii) Rotation through an angle $\pi /4$ about the origin in the anti-clockwise direction.

Prove that the final position of the point is given by the point $(-1/\sqrt{2},7/\sqrt{2})$

Answer 1

$(4,1)\to (1,4)$ by (i)

$(1,4)\to (1+2,4)$ i.e., $(3,4)$ by (ii)

$(3,4)\to 3\cos {45}^o-4\sin {45}^o,3\sin {45}^o+4\cos {45}^{o}or(-\frac{-1}{\sqrt{2}},\frac{7}{\sqrt{2}})$