Equations of Planes — Question 4

Question 4 – Solution

Strategy Move from $P$ along the normal $\vec n=\left\langle 2,-1,2\right\rangle$.

Distance The signed numerator is $2(4)-(-1)+2(5)-3=16$, and $\|\vec n\|=3$, so $\boxed{d=16/3}$.

Foot For $H=P-\lambda\vec n$, impose the plane equation. Since $\lambda=16/9$, $$H=\boxed{(4/9,7/9,13/9)}.$$

Reflection Since $H$ is the midpoint of $PP'$, $P'=2H-P=\boxed{(-28/9,23/9,-19/9)}$.

Verification $H$ satisfies the plane equation, and $P-H$ is parallel to $\vec n$.