# Why is speed measured along the line of sight (LOS) a factor $\sqrt{3}$ smaller for randomly distributed velocities?

I encountered this problem from Introduction to Modern Cosmology by Andrew Liddle.

I had the exact same doubt as the OP of that question had, and the first answer makes sense but I don't understand why "For randomly distributed velocities, the speed V$$_{||}$$ you measure along your line of sight (LOS) is a factor $$\sqrt{3}$$ smaller, i.e. ∼350kms$$^{−1}$$."

How to derive this?

## 1 Answer

The RMS velocity has three components that add in quadrature $$v_{\rm rms}^2 = \sigma_x^2 + \sigma_y^2 + \sigma_z^2\ ,$$ where $$\sigma_i$$ are the RMS velocities along the orthogonal coordinate axes.

If the velocity field is isotropic then we expect the RMS to be the same in any direction and so $$\sigma_x = v_{\rm rms}/\sqrt{3}$$, where the x-axis could be your line of sight.