Finding the coordinate of a point after multiple transformations

Question

Find the new coordinates of $(2,3)$ if following transformations take place: (i) Origin is shifted to $(1,1)$ (ii) Axes are rotated by an angle $45^{\circ}$ in anticlockwise sense (iii) Origin is shifted to $(1,1)$ and then axes are rotated by angle $45^{\circ}$ in clockwise sense (iv) Coordinate axes become $x+y+1=0$ and $x-y+2=0$

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Here is what I have tried:

At first step, coordinates become $(1,2)$, then using $X=x\cos\theta+y\sin\theta, Y=y\cos\theta-x\sin\theta$, we get at the second step $(3/\sqrt{2}, 1/\sqrt{2})$. Now again translating origin to $(1,1)$, and applying rotation, this time $45^{\circ}$ clockwise we get $(1,2-\sqrt{2})$. Finally, changing axes to given lines, we get using the following formula, the point as $(2\sqrt{2}-1,1+1\sqrt{2})$

$$X=\frac{lx+my+n}{\sqrt{l^2+m^2}}\\ Y=\frac{mx-ly+n^{'}}{\sqrt{l^2+m^2}}$$

Turns out the actual answer given is $(1/\sqrt{2}, 6/\sqrt{2})$. Can anybody spot the mistake? Thanks.

Answer 1

(i) Origin is shifted to $(1, 1)$, so coordinates become $(1, 2)$.

(ii) If the axes are rotated by $45^\circ$ anticlockwise then the relation between the original coordinates $(x,y)$ and the new coordinates $(x', y')$ is

$ x = \cos 45^\circ x' - \sin 45^\circ y' $

$ y = \sin 45^\circ x' + \cos 45^\circ y' $

which implies,

$ x' = \cos 45^\circ x + \sin 45^\circ y $

$ y' = -\sin 45^\circ x + \cos 45^\circ y $

Hence, the new coordinates are

$(x', y') = (\dfrac{5}{\sqrt{2}} , \dfrac{1}{\sqrt{2}})$

(iii) In this case, we have a combination of shifting and rotation, so

$(x, y) = (1,1) + R (x', y') $

solving,

$(x', y') =( \cos (-45^\circ) (x-1) + \sin (-45)^\circ (y-1) , -\sin (-45^\circ) (x-1) + \cos (-45^\circ) (y-1)) $

And this evaluates to,

$(x', y') = (-\dfrac{1}{\sqrt{2}}, \dfrac{3}{\sqrt{2}})$

(iv) The point of intersection of the axes is $(-\dfrac{3}{2} , \dfrac{1}{2} )$

The axes orientation is rotated by $45^\circ$ clockwise with respect $xy$ axes.

Hence,

$(x', y') = \cos (-45^\circ) (x+\dfrac{3}{2}) + (\sin -45^\circ) (y-\dfrac{1}{2}) , -\sin( -45^\circ ) (x+\dfrac{3}{2}) + \cos (-45^\circ )(y-\dfrac{1}{2}) ) $

And this evaluates to,

$(x', y') = ( \dfrac{1}{\sqrt{2}} , \dfrac{6}{\sqrt{2}} ) $