1. A particle moves on a given straight line with a constant speed v. at a certain time it is at point P on its straight line path. O is a fixed point. Show that vectorOP × vectorv is independent of the position P.

2. Originally Posted by vritikagupta
1. A particle moves on a given straight line with a constant speed v. at a certain time it is at point P on its straight line path. O is a fixed point. Show that vectorOP × vectorv is independent of the position P.
Every point on the path of the particle may be written:

$\bold{p}=\bold{p_0}+\bold{v}t$,

where $\bold{p_0}$ is the position of the particle at $t=0$.

Then:

$\bold{p}\wedge \bold{v}= (\bold{p_0}+\bold{v}t) \wedge \bold{v}
=\bold{p_0}\wedge \bold{v}+(\bold{v} \wedge \bold{v})t=\bold{p_0}\wedge \bold{v}
$

which is independent of $t$, and so of the particular point $\bold{p}$.

RonL