Line of intersection between two planes

I am stuck on part c) of this question. Here is the link: https://www.imghippo.com/i/KDV1044oc.png
Because planes 1 and planes 2 have line 1 and line 2, I do not see why I can't just use a coordinate from line 1 and a coordinate from line 2 and then find the vector equation. Maybe I have oversimplified the problem.

Plane 1 does not contain l1 and l2. They intersect with the plane at a point. But a bit of simple algebra will give the line l3 which lies in both planes and a sketch may help.

So when line 1 was reflected in plane 1, it produced line 2, which is not in plane 1? So do I find two points on line 1 instead?

So even if a line intersects a line it does not mean that the line lies on the plane. I'm not sure to put this into words but for a line to lie on a plane, does the line need to be lying "flat" on the plane so there is an infinite number of points of intersections between the line and the plane.
Would the direction vector be the direction vector of line 2?
I tried to draw a sketch but I'm not sure if it's accurate:

So even if a line intersects a line it does not mean that the line lies on the plane. I'm not sure to put this into words but for a line to lie on a plane, does the line need to be lying "flat" on the plane so there is an infinite number of points of intersections between the line and the plane.
Would the direction vector be the direction vector of line 2?
I tried to draw a sketch but I'm not sure if it's accurate:

Youre probably suffering from sketching it in 2d. Slightly laboriously you have something like (picture)
https://www.youtube.com/watch?v=qS8MG7jwmc8
Obviously both l1 and l2 line in plane 2 (they define the plane) and so does the intersection point. So what direction vector will lie in plane 1 (think about how the reflection is formed) and fairly trivially in plane 2 as well.

(-2 6 4)?

If thats the direction vector, its not perpendicular to the normal of p1 so the line is not in the plane. But its hard to see what youre thinking without seeing the working.
Rather than just a number (vector), I was expecting you to say something like the desired direction vector is the "average" of the two direction vectors of l1 and l2. It assume the direction vectors have the same magnitude (rescale the scalar multiplier), but thats easy to ensure.

I don't think I understand why it would have to be the average...because l1 does not lie on plane 1.

l1 (red) does not lie in plane 1 (yellow) and therefore neither does its reflection l2 (blue)

However, the black (~horizontal) arrow does lie in the plane and it is perpendicular to the planes normal(s) (black vertical arrows) and meets the vertical reflection line segment in the middle. Meeting it in the middle means the black direction vector is the average of the red and blue ones, assuming they have the same magnitude (hence the previous comment about rescaling mu and the direction vector(s) as appropriate).

I took the average of direction vectors of lines 1 and 2 then multiplied by 2 to give (18 3 7). And the point of intersection between lines 1 and 2 is (-7 2 -7) so the vector equation would be r = (-7 2 -7) + t(18 3 7). I understand that the parallel component lies in plane 1, but how are you certain that it also lies in plane 2?

I took the average of direction vectors of lines 1 and 2 then multiplied by 2 to give (18 3 7). And the point of intersection between lines 1 and 2 is (-7 2 -7) so the vector equation would be r = (-7 2 -7) + t(18 3 7). I understand that the parallel component lies in plane 1, but how are you certain that it also lies in plane 2?

Can you say clearly what the full working for the diretion vector is? (18 3 7) is not perpendicular to the planes normal.
But if you have two lines l1 and l2 lying in a plane then so will a combination like
l3 = 0.5*l1 + 0.5*l2
as the (-7 2 -7) will be unchanged and the new direction vector is 1/2 l1 (in the plane) + 1/2 l2 (also in the plane).

Oh. When you said the average I literally found the average of the direction vectors of l1 snd l2 (eg the x-component of the direction vector of line 1 is 1, and for line 2 it is 10, so the average of 1 and 10 is 5.5 etc...)

Wbf, can you just post your working so
l1 = ..
l2 = ...
l3 = ...