Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2002 / Paper 1 / 1 / b / iii

This plane contains the direction $\mathbf{b}-\mathbf{a}$ and is parallel to $\mathbf{c}$. Put $\mathbf{N}=(\mathbf{b}-\mathbf{a})\times \mathbf{c}$. Its scalar triple product with $\mathbf{a}$ is again $D\ne 0$, proving $\mathbf{N}\ne 0$. HenceThe dot product with $\mathbf{c}$ vanishes, while the dot products with $\mathbf{a}$ and $\mathbf{b}$ equal $\mu$ after normalization. These verify both incidence and parallelism.