Past exam of the mathematics course of the University of Cambridge / 2017 / ia / Paper 3 / 4B / a

Solution

Use the Einstein summation convention. Substitute the given basis transformation into the expansion of the vector:

v_{j} e_{j} = v_{i}^{'} R_{ij} e_{j} .

v_{j} = R_{ij} v_{i}^{'}

. In matrix notation

v = R^{T} v^{'}

R

gives

v_{i}^{'} = R_{ij} v_{j}, v_{i} = R_{ji} v_{j}^{'} .

The placement of the transpose follows from the specified convention

e_{i}^{'} = R_{ij} e_{j}

. It must not be guessed from a different convention for an active rotation of a vector.

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