Equations of Lines — Question 7

Question 7

Reflect the point $P=(4,0,1)$ across the line $L:\vec r=\left\langle 1,1,1\right\rangle+t\left\langle 1,-1,0\right\rangle$. That is, find $P'$ such that $L$ perpendicularly bisects $\overline{PP'}$.

Question 7 – Solution

Strategy First find the perpendicular foot $H$ on the line; then use $H=(P+P')/2$.

Projection point Write $H=(1+t,1-t,1)$. The condition $(P-H)\cdot\left\langle 1,-1,0\right\rangle=0$ gives $$\left\langle 3-t,t-1,0\right\rangle\cdot\left\langle 1,-1,0\right\rangle=4-2t=0,$$ so $t=2$ and $H=(3,-1,1)$.

Reflection Therefore $P'=2H-P=\boxed{(2,-2,1)}$.

Verification The midpoint of $P$ and $P'$ is $H$, and $P-H=\left\langle 1,1,0\right\rangle$ is perpendicular to the direction of $L$.