N2018 P1 Q6

Vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) are such that \(\mathbf{a}\ne \mathbf{0}\) and \(\mathbf{a}\times 3\mathbf{b}=2\mathbf{a}\times \mathbf{c}\).

1. Show that \(3\mathbf{b}-2\mathbf{c}=\lambda \mathbf{a}\), where \(\lambda \) is a constant.[2]
2. It is now given that \(\mathbf{a}\) and \(\mathbf{c}\) are unit vectors, that the modulus of \(\mathbf{b}\) is \(4\) and that the angle between \(\mathbf{b}\) and \(\mathbf{c}\) is \(60{}^\circ \). Using a suitable scalar product, find exactly the two possible values of \(\lambda \).[5]