Vector reflection relative to a unit vector

Question

I am reading a book by Eric Lengyel, Foundations of Game Development - Mathematics, and I am stumped on a particular statement:

"Suppose the component v_⊥a represents everything that is perpendicular to a, we are actually negating a two-dimensional subspace by performing two reflections aligned to vectors that are both orthogonal to a and each other." pp. 66

I am not seeing how this differs from reflecting parallel to a, which he says that negates only a one-dimensional subspace. I am also not sure what vectors he is referring to when he says "each other" at the end of that statement.

As far as I am concerned, they are both just reflecting once, and negating only one component vector. The only difference I see is the axis that is reflected over.

What does that statement mean? How should I look at this?

Answer 1

The author is talking about constructing transform matrices. The first one flips any vector vertically through the horizontal plane perpendicular to $\vec a$, this sort of thing:

The second one flips any vector horizontally (or rather, radially) through $\vec a$ itself (see images in text). Something like this, but all around the vertical axis, in a circle that goes into and out of the the screen:

"I am also not sure what vectors he is referring to when he says "each other" at the end of that statement."

In the images (in the text), only one $\vec v_{\perp \vec a}$ is shown, but if you consider every possible vector, then there's a whole plane of them. Pick any two orthogonal vectors in that plane.