Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics IV / 1 / b

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- Solution b

Solution

Expand the first component using the definition of the cross product:

[a \times (b \times c)]_{1} = a_{2} (b_{1} c_{2} - b_{2} c_{1}) - a_{3} (b_{3} c_{1} - b_{1} c_{3}) = b_{1} (a_{2} c_{2} + a_{3} c_{3}) - c_{1} (a_{2} b_{2} + a_{3} b_{3}) = b_{1} (a \cdot c) - c_{1} (a \cdot b) .

The added

a_{1} b_{1} c_{1}

terms cancel in the last step. Cyclically permuting the component indices gives the other two components, proving the vector triple product identity

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

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