Two particles P P\,P and Q Q\,Q move with constant velocities on a horizontal surface. At time t=0t=0t=0, their position vectors relative to a fixed origin are 0\mathbf{0}0 m and (10i+8j)(10\mathbf{i}+8\mathbf{j})(10i+8j) m respectively. Their velocities are (4i+j)(4\mathbf{i}+\mathbf{j})(4i+j) m s−1^{-1}−1 and (−i−2j)(-\mathbf{i}-2\mathbf{j})(−i−2j) m s−1^{-1}−1 respectively.
Show that the position vector of Q Q\,Q relative to P P\,P at time t t\,t seconds is [(10−5t)i+(8−3t)j]\left[(10-5t)\mathbf{i}+(8-3t)\mathbf{j}\right][(10−5t)i+(8−3t)j] m.
Show that the particles do not collide.
Find the time at which the particles are closest together and find their minimum separation.