Need Help with HW Problem

Question

https://imgur.com/a/W3bCpeP [Link to problem and my solution thus far]

I am pretty sure I found the distance correctly. For the second part, although I think it's right, I'm hesitant because it doesn't feel very mathematically sound--I don't really understand why I'm doing what I'm doing, so if somebody can explain the math behind it or propose a better method of solving, that'd be cool. For the final part, I'm lost, but it seems similar to the previous part, which I don't really understand either.

Thanks.

Answer 1

Hi John - here's a hint.

There's a point, a plane, and the origin. The origin will either 1) lie on the plane 2) not lie on the plane and on the same side as the point or 3) not on the plane, on a side opposite of the point. We can quickly rule out (1) (Is it clear why and how?).

Let's assume at (2) is true, so the point and the origin lie on the same side. Is there anything useful we can say mathematically about this? Remember that this is a problem about finding distances. So perhaps you can state a mathematical property that must be true about distances, assuming that (2) is true.