Question: if p and q are two distinct points in space, show that the set of points equidistant from p and q form a plane.
Work Done:
Note: I'm pretty sure this can be done with vectors and cross products, but in any case this is what I did: I believe I can solve it this way, but it looks quite tedious.
My idea is that I can make a triangle with the origin, p, and the point below/above p on the x / y plane (z component = 0). I do the same with q, and thus have to triangles. Then, I can note the angle between the two triangles by making a triangle with points p, q, and the origin. From the angles I have so far, I believe that I'd be able to find the angle of the plane formed (I think I can also do this through cross product?) and then would be able to rotate the plane extended through the line r(t) according to this angle. However, I can't imagine that this is the way we're wanted to do this… Is the cross product / unit vector idea correct? Thanks.
Best Answer
Just expand $$\|x-p\|^2=\|x-q\|^2$$ to achieve an equation of the plane nearly instantaneously.