1. The problem statement, all variables and given/known data
I am having difficulty understanding the very first step of the following solved problem (I understand the rest of the solution).

How did they obtain the expressions for ##\hat{n}## (the direction of polarization), and ##\hat{k}## (the unit vector pointing in the direction of the wave vector)?

What technique did they use to find the expression ##\frac{1}{\sqrt{6}} (\hat{x}+2\hat{y}+\hat{z})## for the unit vector perpendicular to ##x+y+z=0## plane?

Likewise, how did they get the expression ##\frac{1}{\sqrt{5}} (\hat{y}-2 \hat{z})## for the unit vector parallel to the y-z plane?

I could not find any explanations in my Linear Algebra textbook. So any explanation would be greatly appreciated.

Well that looks strange to me. If the wavevector should be perpendicular to ##x+y+z=0## plane then this plane must be parallel to the planes of constant phase ##\mathbf{k} \cdot \mathbf{r}=C## with ##C## a constant, in fact this plane is one of them. Which means any plane with equation ##x+y+z=C## is traversed by the beam perpendicularly, and we see the possible unit vector of ##k## that that can form such equation by the dot product with ##\mathbf{r}## must subtend the same angle with all three axes.
But I would like to hear the other's opinion.