If we have any two orthonormal vectors A and B in R^2 and we wish to describe the circle they create under rigid rotation (i.e. they rotate at a fixed point and their length is preserved), how can we describe any point along this (unit) circle using a linear combination of A and B? I was thinking that it would be something along the lines of A*cos(θ) + B*sin(θ), but I'm not too sure, for example why not use A*sin(θ)+B*cos(θ). Regardless, I know that any point along this circle can be found because A and B are linearly independent and span all of R^2. I suppose what I'm really interested in, is computations that restrict to this "internal frame", this unit circle (not necessarily centered at(0,0)). I feel this is very much related to the idea that a rotation matrix like [cosθ, -sinθ ; sinθ , cosθ] can rotation a pair of numbers (x,y) to a new pair (x',y') my treating (x,y) as a vector and applying the matrix. At the same time though, this isn't quite my problem; I'm not starting with anything and then rotating it; I have a basis and want to construct a vector. Actually, my full problem (too long to describe here) is embedded in R^3 but this is a subproblem restricted to a 2-dimensional space spanned by A and B.