Consider the usual $(x,y)$ coordinate system and also the $(x^\prime,y^\prime)$ coordinate system, which is rotated $30^\circ$ counterclockwise relative to the $(x,y)$ coordinate system. Let $P$ be the point $(10,10)$ in $(x,y)$ coordinates. What are the coordinates of $P$ in the $(x^\prime,y^\prime)$ coordinate system?

Consider the usual $(x,y)$ coordinate system and also the $(x^\prime, y^\prime)$ coordinate system, which is rotated $45^\circ$ counterclockwise relative to the $(x,y)$ coordinate system. Let $Q$ be the point $(10,6)$ in $(x,y)$ coordinates. What are the coordinates of $Q$ in the $(x^\prime,y^\prime)$ coordinate system?