Find the point $A'$ which results from reflecting $A$ through $L$.

Question

Given that $A(1, 2, 3)$ and line $L$ consists of all the points of the form $(1,2,1)+t(1,-1,1)$

Find the point $A'$ which results from reflecting $A$ through $L$.

Since I know the point on $L$ closest to $A$ I know I should find the distance between them. However, I don't know how reflecting $A$ through $L$ works with linear algebra.

Answer 1

Let $A'(x,y,z).$

Now, solve the following system. $$\frac{x+1}{2}=1+t,$$ $$\frac{y+2}{2}=2-t,$$ $$\frac{z+3}{2}=1+t$$ and $$1\cdot(x-1)-1\cdot(y-2)+1\cdot(z-3)=0.$$