Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 14 / b / Solution

Choose any nonzero auxiliary vector $\mathbf{q}$ not collinear with $\mathbf{p}$. Then $\mathbf{p},\mathbf{q},\mathbf{p}\times \mathbf{q}$ form a basis, so writeTaking the dot product with $\mathbf{p}$ gives $\alpha |\mathbf{p}{|}^2+\beta \mathbf{p}\cdot \mathbf{q}=k$. This separates the two cases requested in the hint.

If $\mathbf{p}\cdot \mathbf{q}\ne 0$, choose $\alpha ,\gamma$ freely and solve for $\beta$:If $\mathbf{p}\cdot \mathbf{q}=0$, the coefficient of $\mathbf{p}$ is fixed and the other two are free:These formulas describe the whole plane with normal $\mathbf{p}$, including $k=0$. Completeness follows from the basis expansion, not only from checking that the displayed vectors satisfy the equation. The parametrization of a plane with an auxiliary vector also gives a single formula valid in both cases.