Unit Vectors for Polarization and Wave Vector Directions

[roam]

Homework Statement

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)?

Homework Equations

$k=\frac{\omega}{\lambda f} = \frac{\omega}{c}=\frac{2 \pi}{\lambda}$

$E(r, t) = E_0 \ cos (k.r - \omega t) \hat{n}$

$B(r,t) = \frac{1}{c} E_0 \ cos (k.r - \omega t) (\hat{k} \times \hat{n})$

The Attempt at a Solution

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.

 

[blue_leaf77]

Where do you get that solution from?

 

[roam]

Where do you get that solution from?

This was the solution provided by my teacher. I don't understand, where did he get he get the expressions for $\hat{n}$ and $\hat{k}$ from?

 

[blue_leaf77]

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.