2023 HCI J2 MYE P2 Q3

1. Given that \(\mathbf{u}\), \(\mathbf{v}\) and \(\mathbf{w}\) are nonzero vectors such that \(\mathbf{u}=\mathbf{v}\times \mathbf{w}\), state the geometric relationship that \(\mathbf{u}\) has with \(\mathbf{v}\) and \(\mathbf{w}\) respectively.[1]

The vector \(\mathbf{r}\) is given as \(\mathbf{r}=\mathbf{a}\times \left( \mathbf{b}\times \mathbf{c} \right)\), where \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) are non-zero, non-parallel vectors.

2. Explain why \(\mathbf{r}\) can be expressed as \(\mathbf{r}=\lambda \mathbf{b}+\mu \mathbf{c}\), for constants \(\lambda \) and \(\mu \).[2]
3. Find the value of \(\lambda \mathbf{a}\cdot \mathbf{b}+\mu \mathbf{a}\cdot \mathbf{c}\).[2]