(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A flashlight located at the origin (0, 0, 0) shines a beam of light towards a flat mirror. The beam reflects off of the mirror at (8, 4, 1) and then passes through (10, 8, 5). What is the equation of the plane that contains the mirror?

2. Relevant equations

The only assumed knowledge is what has been covered in this specific chapter; three-dimensional Cartesian coordinates, vectors, dot product, cross product, and finding the equations of lines and planes in space.

3. The attempt at a solution

I know that to find the equation of a plane one needs:

* a point on the plane, and

* a vectorn, perpendicular to the plane.

The point where the light strikes the mirror satisfies the first requirement, so our point is P(8, 4, 1). Finding a vector normal to the plane (mirror) is where I'm stumped. I found the vector created by the reflecting light, (10-8, 8-4, 5-1) = (2, 4, 4). I initially tried to use the cross product with vector (8, 4, 1) and vector (2, 4, 4). I know that gives me a vector perpendicular to both, but I quickly realized that I don't have any way (or don't know how) to find the orientation of that vector to the plane I'm trying to find. So it doesn't really help.

Someone suggested that the normal vector to the mirror is the bisector vector of (-8, -4, -1) and (2, 4, 4). While I imagine this would work, I'm almost certain that is not the approach we are intended to use. We haven't covered that, and I'm pretty sure it's not assumed knowledge at this point.

I also tried all types of other stuff, but the truth is that I was just getting desperate and trying everything I could think of (without success).

Any help would be much appreciated.

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# Homework Help: Equation of a Plane

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