How to reflect a vector across a rotated plane?

Question

Let's say I have vector $\vec{A}$ which has points (-5,6,3) and (4,-2,4). I also have plane $P$ which is defined by the three points, (4,-2,4), (5,-3,4), and (4,-3,4). My question is, how would I reflect $\vec{A}$ across plane $P$?

Answer 1

Hint 1 First try and find two vectors which span your plane $P$. This is done by taking the difference between pairs of points in $P$ which are not co-linear.

Hint 2 Find a vector which is orthogonal to the plane $P$. Have you met the cross product yet?

Hint 3 Find the distance between a point of $\vec{A}$ and the plane $P$ along the vector orthogonal to $P$. Do this for another point. What would it mean geometrically to reflect these points across the plane $P$?