Question

A particle moves on a given straight line with a constant speed v. At a certain time it is at a point P on its straight line path. O is a fixed point. Show that $\overrightarrow{OP}\times \overrightarrow{v}$ is independent of the position P.

Answer 1

The particle moves on the striaght line PP' at speed v.

From the figure,

$\overrightarrow{OP}\times v=\left(OP\right)v\sin \theta \hat{n}=v\left(OP\right)\sin \theta \hat{n}=v\left(OQ\right)\hat{n}$

It can be seen from the figure, $OQ=OP\sin \theta =O{P}^{\prime }\sin {\theta }^{\prime }$

So, whatever may be the position of the particle, the magnitude and direction of $\overrightarrow{OP}\times \overrightarrow{v}$ remain constant.

$\therefore \overrightarrow{OP}\times \overrightarrow{v}$ is independent of the position P.