A point object is moving with a speed $\nu$ before an arrangement of two mirrors as shown in figure. Find the magnitude of velocity of image in mirror $M_1$ with respect to image in mirror $M_2$.

We have to find magnitude of relative velocity between ${\nu }_{I_1}$ and ${\nu }_{I_2}$. The angle between these two vectors is $2\theta$.

hence $$\begin{gathered} {\nu }_r = {\nu }_{I_1}-{\nu }_{I_2} \\ \therefore \left|{\nu }_r\right| =\left| {\nu }_{I_1}-{\nu }_{I_2}\right| \end{gathered}$$

There is nothing but magnitude of subtraction of two velocity vectors of equal magnitudes $\nu$ each and angle between them equal to $2\theta$.

hence, $\left|{\nu }_r\right|=\sqrt{{\nu }^2+{\nu }^2-2\left(\nu \right)\left(\nu \right)\cos 2\theta }$

Solving these two equations , we get

$\left|{\nu }_2\right|=2v\sin \theta$