Inertial frame where plane waves have the same frequency

[Halleluwah]

Homework Statement

Plane harmonic waves of $1/p, 1/q, 1/r$ and $1/s$ are travelling, respectively, in the directions of the (non-unit) vectors $(1,1,1), (1,-1,-1), (-1,1,-1)$ and $(-1,-1,1)$. Show that there exists an inertial coordinate system in which they have the same frequency if and only if
$$3(p+q-r-s)^2 + 3(p-q+r-s)^2 + 3(p-q-r+s)^2 < (p+q+r+s)^2$$

Homework Equations

The Attempt at a Solution

So I can make null forward pointing vectors which are:
$$1/p(1,1/\sqrt{3}, 1/\sqrt{3},1/\sqrt{3})$$
and so on.
I notice each of the vectors components 'correspond' to one of the terms in the inequality. That is, the right hand side corresponds to the first compenent of each vector, the second component of the vectors corresponds to the first term on the left, the third component of the vectors corresponds to the second term on the left, and the fourth component of the vector corresponds to the third term on the left.

I would like some kind of hint because I'm not sure where to go with this. It seems that I am not seeing a key fact.

 

[Dazed&Confused]

I made a mistake in copying this out. It should be ''plane harmonic waves of frequencies.''