Past exam of the mathematics course of the University of Cambridge / 2016 / ia / Paper 1 / 5A / a

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- Solution a

Solution

Use suffix notation with indices ranging from

1

to

3

, and sum over repeated indices. By the component definitions of the dot product and cross product,

a \cdot (b \times c) = a_{i} ϵ_{ijk} b_{j} c_{k} .

The Levi-Civita symbol is unchanged by a cyclic permutation of its three indices: the cycle is two transpositions and hence has positive sign. Thus

ϵ_{ijk} = ϵ_{kij}

. Commuting the scalar components gives

a_{i} ϵ_{ijk} b_{j} c_{k} = c_{k} ϵ_{kij} a_{i} b_{j} = c_{k} (a \times b)_{k} .

Therefore the scalar triple product obeys

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

Cyclically moving the vectors preserves the scalar triple product; interchanging just two reverses its sign.

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