 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/a/solution

r \cdot \hat{u}

\hat{u}

. Multiplying by that unit vector makes

P_{\hat{u}} (r)

r

onto its span.

For the other expression, the vector triple product identity gives

R_{\hat{u}} (r) = r (\hat{u} \cdot \hat{u}) - \hat{u} (\hat{u} \cdot r) = r - \hat{u} (r \cdot \hat{u}) .

\hat{u}

through the origin. Its dot product with

\hat{u}

is zero, while the difference from

r

is parallel to

\hat{u}

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