4 A particle moves on a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular. At time \(t\), the particle's position vector, \(\mathbf { r }\), is given by $$\mathbf { r } = 4 \cos 3 t \mathbf { i } - 4 \sin 3 t \mathbf { j }$$

1. Prove that the particle is moving on a circle, which has its centre at the origin.
2. Find an expression for the velocity of the particle at time \(t\).
3. Find an expression for the acceleration of the particle at time \(t\).
4. The acceleration of the particle can be written as $$\mathbf { a } = k \mathbf { r }$$ where \(k\) is a constant. Find the value of \(k\).
5. State the direction of the acceleration of the particle.