Question

Suppose you pick a random point anywhere in space. Then you create 3 orthogonal vectors from that point. Using these vectors, you create 3 orthogonal planes that all intersect at this point. Now, if you reflect everything through the xy, xz, and yz planes in any order is that the same as simply just taking our original point and from there creating a line to any arbitrary point and sliding that point along the line until you're on the other side of the point we picked by the same distance, and then repeating that for every point?

Answer 1

To clarify, I'm saying we reflect everything by each of the three mirroring planes once, it's just that the order in which we reflect that I don't think matters.

Also, for the second transformation, what i mean is something like this:

that black center point is where everything gets inverted through, so that these two corners are swapped.

Answer 2

Uhhh... Well I just figured it out never mind lol. It's clearly true if you just think of matrices of transformation.