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 (OP) ⃗×v ⃗ is independent of the position P.

Sol. The particle moves on the straight line PP’ at speed v. From the figure, (OP) ⃗ v = (OP)v sin θ n ̂ = v(OP) sin θ n ̂ = v(OQ) n ̂ It can be seen from the figure, OQ = OP sin θ = OP’ sin θ’ So, whatever may be the position of the particle, the magnitude and direction of (OP) ⃗ v ⃗ remain constant. ∴ (OP) ⃗ x v ⃗ is independent of the position P.