so I'm working on a past paper for my maths exam and there are no provided answers. I am given two planes $2x + 4y -4z = 22$ and $\begin{pmatrix}-1\\3\\2\end{pmatrix} + s\begin{pmatrix}1\\-1\\-1\end{pmatrix} + t\begin{pmatrix}1\\0\\0\end{pmatrix}$ and I have to reflect the 2nd in the 1st.
I also have the line of intersection from a previous part of a question but I'm not sure if that's useful here.
My idea is to just reflect 3 points and connect them to form a plane but I was wondering if there's an easier way to do this. Any help is very much appreciated!
Sounds like one of the easiest ways to do this to me. In fact, you can just apply the reflection formula to the expression that you have for an arbitrary point on the second plane. The algebra can get a bit messy, so if you want to minimize the chance of errors, you can reflect those three vectors individually. Be careful, though: the two direction vectors in that expression need to be handled a bit differently from the fixed point on the plane.
Since you’ve already computed the intersection $L$ of the two planes, you might have an easier way. $L$ lies on the reflecting plane, so it is fixed by the reflection—it also lies on the reflected image of the first plane (in fact, it’s the intersection of that plane with its reflection). So, you just need one more point to define the reflected image. Take any point on the first plane not on $L$ and compute its reflection, then work out the resulting plane.