I need to find the reflection of point $P(1,2,3)$ w.r.t line mirror $(x-1)/2 =(y-1)/3 = (z+1)/1$
I know one method to do it i.e by first finding the foot of perpendicular of P on the line by using direction ratios and solving with line and then when the coordinates of foot of perpendicular are known ,I can simply use mid point formula to get the coordinates of reqd reflection point.
I want to know is there any other /alternative method to solve this question ,that'll help me alot.
Any help appreciated!
The point on the given line it's $C(1+2t,1+3t,-1+t)$ and the needed point it's $B(1+4t,6t,-5+2t),$ where $C$ is a midpoint of $PB$.
Thus, $$(2t,3t-1,-4+t)(2,3,1)=0,$$ which gives $$4t+9t-3-4+t=0$$ or $t=\frac{1}{2}$ and we got that $B(3,3,-4).$
Actually. We did not find coordinates of the foot, but coordinates of $C$, which one step less, than in your solution.