 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-2/11/d/i/solution

Take the cross product with

\hat{q}

r \times \hat{q} = p \times \hat{q} + t (\hat{q} \times \hat{q}) = p \times \hat{q} .

Thus one choice is

\hat{a} = \hat{q}, c = p \times \hat{q} .

Conversely, this cross product equation implies

(r - p) \times \hat{q} = 0

, so

r - p

is parallel to

\hat{q}

. It therefore describes precisely the original straight line.

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