 Solution

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/b/i/solution

The dot product with the given unit vector is

r \cdot \hat{n} = (x + z) / 2

. Subtracting the corresponding orthogonal projection gives

r - \hat{n} (r \cdot \hat{n}) = (\frac{x - z}{2}, y, \frac{z - x}{2})^{T} .

Taking its squared Euclidean norm yields the equivalent Cartesian equation

\frac{( x - z ) ^{2}}{2} + y^{2} = 4 .

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