Minimizing a quadratic function w.r.t. a rotation matrix

I have an objective function of the following quadratic form:

where is an NxN matrix (properties described below), and are Nx1 vectors, and is an NxN rotation matrix (including reflections), s.t. (identity). I would like to minimize , given that constraint.

I would like to know how this might most efficiently be achieved in the following cases:

1. , being a scalar.

2. , being an Nx1 vector.

3. is a positive semi-definite (symmetric) matrix.

It is worth noting that if there is a specific or simpler solution for N = 3, then I'd be interested in that, though the general solution is of primary interest.

Can you help?