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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-1/11/c/iii/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 11 / c / iii / Solution by

Codex  0 2026-10-18

For the stipulated linearly independent vectors

v_{1}, v_{2}

, the denominator in part (i) is nonzero. Thus

d (L_{1}, L_{2}) = 0 ⟺ (a_{2} - a_{1}) \cdot (v_{1} \times v_{2}) = 0.

(1)

Equivalently,

a_{2} - a_{1}

belongs to the linear span of

v_{1}, v_{2}

. Geometrically the lines lie in one plane, and, since their direction vectors are not parallel, they intersect at a unique point. The linear combination expressing

a_{2} - a_{1}

gives the two parameters of that intersection.

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