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:
$\displaystyle \bold{p}=\bold{p_0}+\bold{v}t$,
where $\displaystyle \bold{p_0}$ is the position of the particle at $\displaystyle t=0$.
Then:
$\displaystyle \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 $\displaystyle t$, and so of the particular point $\displaystyle \bold{p}$.
RonL