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 $\left(2-\frac1{\sqrt2},1+\frac3{\sqrt2}\right)$.