"With a 2 dimensional surface: We take (2, 1) as the center point and consider a transformation with a rotation angle of 45◦ 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.
Thank you!
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Draw out the graph. Also draw out the situation when it is rotated $90°$, calculate the gradient of line connecting point (0,2) and (3,3), and then derive the gradient of its perpendicular line, which pass through the desired point. Finally use the gradient with the length between the centre and the desired point to obtain the coordinate of the desired point. The approach is quite tedious though..
Hint (assuming that you are working on $\mathbb{R}^2$). Write the $2\times 2$ matrix $R$ of the rotation of $45°$ about the origin. Let $\mathbf{c}=(2,1)\in \mathbb{R}^2$ the coordinates of the centre of the rotation. The transformation you are looking for is $$ \mathbf{x} \mapsto R(\mathbf{x}-\mathbf{c}) + \mathbf{c}. $$