there exists an orthogonal transformation such that

Question

Prove or Disprove: There exists an orthogonal transformation T: $\mathbb{R}^3 \to \mathbb{R}^3 $ such that $$ T\left(\begin{bmatrix} 2 \\ 3 \\ 0 \\ \end{bmatrix}\right)= \begin{bmatrix} 3 \\ 0 \\ 2 \\ \end{bmatrix} and T\left(\begin{bmatrix} -3 \\ 2 \\ 0 \\ \end{bmatrix}\right)= \begin{bmatrix} 2 \\ -3 \\ 0 \\ \end{bmatrix} $$

I am not quite sure how to begin this problem. All I know is that a transformation T: $\mathbb{R}^n \to \mathbb{R}^n $ is orthogonal if T preserves length. Not quite sure how to proceed from there.

Thanks!

Answer 1

If only you knew something about the angle between the vectors $(2,3,0)$ and $(-3,2,0)$ and then also about the angle between their images...