# Rotation problem [duplicate]

With a 2D surface, we take $$(2, 1)$$ as the center point and consider a transformation with a rotation angle of $$45^\circ$$ so point $$(3, 3)$$ is transformed into point?

I'm really close to getting the answer! I've gotten $$(-1/\sqrt2,3/\sqrt2)$$ but the answer is $$(2-1/\sqrt2, 3+1/\sqrt2)$$. Please tell me what I'm missing.

The displacement vector from $$(2,1)$$ to $$(3,3)$$ is $$(1,2)$$. Rotated counterclockwise by $$45^\circ$$, $$(1,2)$$ becomes $$\begin{bmatrix} \cos45^\circ&-\sin45^\circ\\ \sin45^\circ&\cos45^\circ \end{bmatrix}\begin{bmatrix}1\\2\end{bmatrix}$$ $$=\frac1{\sqrt2}\begin{bmatrix} 1&-1\\1&1\end{bmatrix}\begin{bmatrix}1\\2\end{bmatrix}=\frac1{\sqrt2}\begin{bmatrix}-1\\3\end{bmatrix}$$ Thus $$(3,3)$$ is transformed to $$(2,1)+\frac1{\sqrt2}(-1,3)$$ or $$\left(2-\frac1{\sqrt2},1+\frac3{\sqrt2}\right)$$.