7. Relative to a fixed origin \(O\), the points \(X\) and \(Y\) have position vectors \(( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m }\) and \(( 12 \mathbf { i } + \mathbf { j } ) \mathbf { m }\) respectively, where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.

1. Find the distance \(X Y\). A particle \(P\) of mass 2 kg moves from \(X\) to \(Y\) in 4 seconds, in a straight line at a constant speed.
2. Show that the velocity vector of \(P\) is \(( 2 \mathbf { i } + 1 \cdot 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). The particle continues beyond \(Y\) with the same constant velocity.
3. Write down an expression for the position vector of \(P t\) seconds after leaving \(X\).
4. Find the value of \(t\) when \(P\) is at the point with position vector \(( 16 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m }\). When it is moving with the same constant speed, \(P\) collides directly with another particle \(Q\), of mass 4 kg , which is at rest. \(P\) and \(Q\) coalesce and move together as a single particle.
5. Find the velocity vector of the combined particle after the collision.
   (5 marks)