 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1/14/d/solution

The squared Euclidean distance along the line is

∣ a + t d ∣^{2} = ∣ a ∣^{2} + 2 t (a \cdot d) + t^{2} ∣ d ∣^{2}, d = b \times c .

Its minimum occurs at

t = 0

exactly when its derivative there vanishes, so $a \cdot (b \times c) = 0$ . Since

b, c

are independent, their linear span is the two-dimensional orthogonal complement of

b \times c

. Consequently

a = α b + β c

for unique real

α, β

∣ d ∣^{2}

proves that the stationary distance is a strict global minimum.

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