Find the point symmetric to M(-1,0,-1) with respect to the…

Find the point symmetric to $M(-1,0,-1)$ with respect to the plane $2x+y-z+7=0$.

Explanation of Method

The mirror image of the point M(−1, 0, −1) in a plane.

$M'(-3,-1,0)$ (Problem 259) is the midpoint of $MM^{*}$: $M^{*}=2M'-M=(-5,-2,1)$.

Check: $M$ and $M^{*}$ are both at distance $\sqrt6$ from the plane, and $\overrightarrow{MM^{*}}=-2\{2,1,-1\}$.

$M$ $M'$ $M^{*}$

$M'(-3,-1,0)$ is the projection of $M$ onto the plane (Problem 259) and the midpoint of $MM^{*}$, with $M^{*}(-5,-2,1)$.

\[\boxed{\,M^{*}=2M'-M=(-5,-2,1)\,}\]