7. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane] A small smooth ball of mass \(m\) kg is moving on a smooth horizontal plane and strikes a fixed smooth vertical wall. The plane and the wall intersect in a straight line which is parallel to the vector \(2 \mathbf { i } + \mathbf { j }\). The velocity of the ball immediately before the impact is \(b \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(b\) is positive. The velocity of the ball immediately after the impact is \(a ( \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(a\) is positive.

1. Show that the impulse received by the ball when it strikes the wall is parallel to \(( - \mathbf { i } + 2 \mathbf { j } )\). Find
2. the coefficient of restitution between the ball and the wall,
3. the fraction of the kinetic energy of the ball that is lost due to the impact.