2019 NJC Promo Q9

Non-zero and non-parallel vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) are such that \(\mathbf{b}\times 3\mathbf{c}=\mathbf{c}\times \mathbf{a}\).

1. Determine the relationship between \(\mathbf{c}\) and \(\mathbf{a}+3\mathbf{b}\), justifying your answer.[2]

It is given that \(\mathbf{a}\) and \(3\mathbf{b}\) are unit vectors and that the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(60{}^\circ \).

2. Evaluate \(\left| \mathbf{a}+3\mathbf{b} \right|\).[3]
3. Given further that \(\mathbf{a}+3\mathbf{b}\) makes an angle of \(60{}^\circ \), \(120{}^\circ \) and \(135{}^\circ \) with the positive \(x\)-, \(y\)- and \(z\)-axes respectively, show that \(\mathbf{c}\) is parallel to \(\mathbf{i}-\mathbf{j}-\sqrt{2}\mathbf{k}\).[3]