Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 11

Three vectors form a basis of three-dimensional space exactly when their scalar triple product is nonzero:It is the determinant of the matrix with these columns, so this condition is equivalent to linear independence and spanning the whole space.

Take the dot product of the expansion of $\mathbf{y}$ with $\mathbf{v}\times \mathbf{w}$. The terms containing $\mathbf{v}$ and $\mathbf{w}$ vanish, leaving $\mathbf{y}\cdot (\mathbf{v}\times \mathbf{w})=\alpha \Delta$. Cycling the same argument yieldsThe three cross-product numerators use the reciprocal basis.

Take the cross product with $\mathbf{c}$. Since $\mathbf{c}\times \mathbf{c}=0$, the equation impliesAs $\mathbf{c}\ne 0$, the displacement is parallel to $\mathbf{c}$. ThereforeThis is a necessary condition; substitution must still determine which points on this straight line satisfy the full equation.

Substitute $\mathbf{x}=\mathbf{a}+\lambda \mathbf{c}$ into the original equation and cancel $\mathbf{a}$. Because $\mathbf{c}$ is nonzero, the scalar condition isIf $\mathbf{b}\cdot \mathbf{c}\ne 1$, there is one value, $\lambda =(\mathbf{b}\cdot \mathbf{a})/(1-\mathbf{b}\cdot \mathbf{c})$. If $\mathbf{b}\cdot \mathbf{c}=1$ and $\mathbf{b}\cdot \mathbf{a}=0$, every real $\lambda$ works. If $\mathbf{b}\cdot \mathbf{c}=1$ and $\mathbf{b}\cdot \mathbf{a}\ne 0$, the scalar equation is inconsistent and there is no solution.

Thus “undefined” in the last case means that no parameter value solves the equation; it must not be used to conceal the distinct case with infinitely many solutions.

In the nonsingular case the unique solution isGeometrically this is a single selected point of the straight line through $\mathbf{a}$ parallel to $\mathbf{c}$.

When $\mathbf{b}\cdot \mathbf{c}=1$ and $\mathbf{b}\cdot \mathbf{a}=0$, the solution set is that entire straight line, $\mathbf{a}+\lambda \mathbf{c}$. The remaining case has empty solution set. In matrix terms the equation is $(I-\mathbf{c}{\mathbf{b}}^T)\mathbf{x}=\mathbf{a}$, a rank-one matrix update of the identity matrix.