Past exam of the mathematical methods I course of the University of Oxford / 2013 resit / 10 / b

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- Solution b

Solution

The first operation acts nearest the column vector, so the composite rotation matrix is

M = R_{z} (α) R_{x} (β)

:

M = cos α sin α 0 - sin α cos β cos α cos β sin β sin α sin β - cos α sin β cos β .

At

α = β = π /2

this reduces to

M = 010001100, (x, y, z) \mapsto (z, x, y) .

Solving

M v = v

gives

v_{x} = v_{y} = v_{z}

. Normalization yields the fixed-axis unit vectors

v = \pm \frac{1}{3} (1, 1, 1) .

Both signs describe the same unoriented axis. The operation is a cyclic permutation, and

M^{3} = I

; it is not generally obtained by reversing the multiplication order.

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