Rotation in three dimensions

A rotation in three dimensions is an element of

SO (3)

. It fixes an axis and acts as a two-dimensional rotation on the plane perpendicular to that axis.

Rotation generator ( $J_{k}$ )

In the three-dimensional vector representation, the Hermitian matrices

(J_{k})_{ij} = - i ϵ_{kij}

R (θ, n) = exp (- i θ n^{k} J_{k})

. They satisfy

[J_{i}, J_{j}] = i ϵ_{ijk} J_{k}

. The real antisymmetric matrices

A_{k} = - i J_{k}

are the corresponding Lie algebra generators of

SO (3)

. Their commutator gives the leading change produced by interchanging two infinitesimal rotations.

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