 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2009/paper-1/15/b/solution

(a \times b) \times (a \times c) = a [(a \times b) \cdot c] - c [(a \times b) \cdot a] = [a, b, c] a .

The last inner product vanishes by perpendicularity. Taking the inner product with

d

gives

[a \times b, a \times c, d] = d \cdot [(a \times b) \times (a \times c)] = (a \cdot d) [a, b, c] .

This is the identity for cross products with a common vector, and holds without nondegeneracy assumptions.

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