Linear algebra: Analytic geometry problem.

Question

Let $p$ be a line given by the equation $x-1=\frac{y}{2}=\frac{z+3}{2}$, and $q$ a line with the equation $\frac{x}{2}-1=y-2=\frac{z+1}{2}$. If we reflect the $p$ over the plane $\Pi$ we get the line $q$. What is the equation of $\Pi$ and how many solutions are there?

Answer 1

Start from the other end. Suppose you have one line and a plane. What is the reflection? That depends on how the line and the plane are placed relative to one another. There are three cases to consider:

1. The line intersects the plane in a single point. Then the reflection passes through that same point. The plane spanned by the pair of lines is orthogonal to the reflection plane. And the plane spanned by the pair of lines intersects the reflection plane in a line which is a bisector of the angle formed by the two lines.
2. The line is parallel to the plane. Then the reflection is parallel to the first line. The plane spanned by the pair of lines is again orthogonal to the reflection plane.
3. The line lies fully within the plane. Then the reflection is identical to the first line.
4. There is no other case to consider, the list so far is exhaustive.

Now reversing these, you get

1. If two lines intersect in a single point, then you get two possible planes of reflection, one for each of the two angular bisectors of the angle formed by the two lines.
2. If two lines are parallel, there is exactly one reflection plane, halfway between them, parallel to both and orthogonal to the plane spanned by both.
3. If the lines are identical, then there are infinitely many possible planes, since any plane through that line will satisfy the requirements.
4. If the lines are skew, there is no possible plane of reflection.

I'll leave it to you to have a closer look at your specific lines and work out which of these cases applies.