 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/iii/solution

Decompose

p

\hat{a}

. Since

p \times \hat{a}

removes the parallel component,

∣ c ∣ = ∣ p \times \hat{a} ∣ = ∣ p - (p \cdot \hat{a}) \hat{a} ∣.

The last vector is the displacement to the point of the straight line nearest the origin. Hence

c is the perpendicular distance from the origin to the line .

The original PDF has

c = ∣ c ∣

, not the squared magnitude inserted by the OCR.

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