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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?

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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.

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