Vector analysis:Finding components after rotation

Question

A vector $\vec{a}$ has components $2p$ and $1$ with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counterclockwise sense. If, with respect to the new system, $\vec{a}$ has components $p + 1$ and $1$, then how to find the value of $p$ from this information?

Answer 1

Hint: Length is preserved by rotations.