2024 HCI P2 Q4

1. For any non-parallel and non-zero vectors \(\mathbf{m}\) and \(\mathbf{n}\), explain clearly and show that
   \({{\left( \mathbf{m}\cdot \mathbf{n} \right)}^{2}}+{{\left| \mathbf{m}\times \mathbf{n} \right|}^{2}}={{\left| \mathbf{m} \right|}^{2}}{{\left| \mathbf{n} \right|}^{2}}\).[2]

\(P\) and \(Q\) are two distinct points, where \(\overrightarrow{OP}=\mathbf{p}\) and \(\overrightarrow{OQ}=\mathbf{q}\). It is also known that \(\mathbf{p}\) and \(\mathbf{q}\) are non-zero vectors.
Two non-parallel lines \({{l}_{1}}\) and \({{l}_{2}}\) have vector equations:

\(\mathbf{r}=\mathbf{p}+s\mathbf{u}\),\(s\in \mathbb{R}\) and \(\mathbf{r}=\mathbf{q}+t\mathbf{v}\), \(t\in \mathbb{R}\) respectively.

2. If \(\mathbf{v}\times \left( \mathbf{p}-\mathbf{q} \right)=0\), what can be said about the relationship between the two lines? Justify your answer.[3]