 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2017/9/b/solution

Write both cross products with the Levi-Civita symbol. Its contraction identity gives

[a \times (b \times c)]_{i} = ϵ_{ijk} ϵ_{k l m} a_{j} b_{l} c_{m} = (δ_{i l} δ_{jm} - δ_{im} δ_{j l}) a_{j} b_{l} c_{m} = b_{i} (a \cdot c) - c_{i} (a \cdot b) .

Therefore the vector triple product identity is

a \times (b \times c) = (a \cdot c) b - (a \cdot b) c .

The order matters: the cross product is not associative, and exchanging the two nested arrangements changes the expression.

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