With a two-dimensional, take $(2, 1)$ as the center point and consider a transformation with a rotation angle of $45^\circ$, then point $(3, 3)$

Question

Then point $(3, 3)$ transformed into what point?

I couldn't solve this problem, I need step by step solution to understand and learn. Does it matter two dimensional or three dimensional? What if rotation angle other than $45^\circ$.?

Answer 1

When $(2,1)$ is the center, $(3,3)$ will correspond to the point $(1,2)$.

Now, the tranformation rotates the bector by 45°.

Take the basis vectors $(1,0)$ and $(0,1)$.

$(1,0)$ will be transformed to $v = (-1/ \sqrt{2}, 1/ \sqrt{2})$ and $(0,1)$ will be transformed to $u= (1/ \sqrt{2}, 1/ \sqrt{2})$.

The transformation matrix will be $[v^T:u^T]= A$

So, the point $x = (1,2)$ will be transformed to,

$Ax= (1/ \sqrt{2}, 3/ \sqrt{2})$