Vector perpendicular to two directions shown to be parallel to a vector triple product

The vectors \(\mathbf{p}\) and \(\mathbf{q}\) in three dimensions are non-zero and are not parallel.

The vector \(\mathbf{u}\) is given by \(\mathbf{u}=|\mathbf{q}|^2\,\mathbf{p}-(\mathbf{p}\cdot\mathbf{q})\,\mathbf{q}\), and \(\mathbf{u}\ne\mathbf{0}\).

1. Show that \(\mathbf{u}\) is perpendicular to \(\mathbf{q}\).[2]
2. Let \(\mathbf{n}=\mathbf{p}\times\mathbf{q}\), which is a non-zero vector.
   1. Explain why \(\mathbf{p}\cdot\mathbf{n}=0\) and \(\mathbf{q}\cdot\mathbf{n}=0\).
   2. Show that \(\mathbf{u}\cdot\mathbf{n}=0\).[4]
3. Hence, state how the vectors \(\mathbf{u}\) and \(\mathbf{q}\times\mathbf{n}\) are related geometrically, giving a brief reason.[2]