Find all vectors $(x,y,z)$ whose value under the rotation..

Question

Find all vectors $(x,y,z)$ whose value under the rotation through the angle $\pi/2$ about the $y$-axis is the vector $(z,x,y)$

Can anyone please provide some help in answering this question

Answer 1

Hint: If you rotate $(x,y,z)$ around the $y$-axis by an angle equal to $\frac\pi2$, you will get $(-z,y,x)$. So, solve the equation $(-z,y,x)=(z,x,y)$.

Answer 2

Rotation metrix is,$ \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \\ \end{pmatrix} $

So required condition is$ \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \\ \end{pmatrix} $ $\begin{pmatrix} x \\ y \\ z \\ \end{pmatrix} = \begin{pmatrix} z \\ x \\ y \\ \end{pmatrix}$

This gives, $z=z,y=x,-x=y$

$\therefore x=y=0$.

so any vector of the form $(0,0,z)$ satisfies the required condition.