2020 YIJC P1 Q1

Vectors \(\mathbf{a}\),\(\mathbf{b}\) and \(\mathbf{c}\) are such that \(\mathbf{b}\ne \mathbf{0}\) and \(3\mathbf{a}\times \mathbf{b}=7\mathbf{b}\times \mathbf{c}\).

1. Show that \(3\mathbf{a}+7\mathbf{c}=\lambda \mathbf{b}\), where \(\lambda \) is a constant.[2]
2. It is now given that \(\mathbf{a}\)and \(\mathbf{b}\) are unit vectors, that the modulus of \(\mathbf{c}\) is \(\frac{3}{7}\) and that the angle between \(\mathbf{a}\) and \(\mathbf{c}\) is \(60{}^\circ \). Using a suitable scalar product, find exactly the two possible values of \(\lambda \).[4]