Two particles P P\,P and Q Q\,Q are moving in separate straight lines across a smooth horizontal surface.
P P\,P moves with constant velocity (3i+4j)(3\mathbf{i} + 4\mathbf{j})(3i+4j) m s−1^{-1}−1.
Q Q\,Q moves from the point with position vector (2i−5j)(2\mathbf{i} - 5\mathbf{j})(2i−5j) m to the point with position vector (20i+19j)(20\mathbf{i} + 19\mathbf{j})(20i+19j) m during a 3 second period.
Show that P P\,P and Q Q\,Q move along parallel lines.
Stevie says that Q Q\,Q is also moving with a constant velocity of (6i+8j)(6\mathbf{i} + 8\mathbf{j})(6i+8j) m s−1^{-1}−1. Explain why Stevie may be incorrect.
A third particle R R\,R moves with constant speed 5 m s−1^{-1}−1, in a straight line, across the same surface. P P\,P and R R\,R move along lines that intersect at a fixed point XXX. It is given that P P\,P passes through X X\,X exactly 2 seconds after R R\,R passes through XXX, and that P P\,P and R R\,R are exactly 510 5\sqrt{10}\,510 metres apart 3 seconds after R R\,R passes through XXX. Show that P P\,P and R R\,R move along perpendicular lines.