2024 JPJC Vectors Tutorial Q11

The vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) are non-zero, non-parallel vectors. If \(\mathbf{a}\times \mathbf{b}=\mathbf{a}\times 3\mathbf{c}\), then show that \(\mathbf{a}=k\left( \mathbf{b}-3\mathbf{c} \right)\) for some constant \(k\).[3]

Given that \(\mathbf{u}\) and \(\mathbf{v}\) are non-zero vectors such that \(\mathbf{u}\times \mathbf{v}=\left( \mathbf{u}-\mathbf{v} \right)\times \left( \mathbf{u}+\mathbf{v} \right)\), what can be deduced about \(\mathbf{u}\) and \(\mathbf{v}\)?[2]