Can a vector $u$ be always written as $e_{i}\Vert v\Vert \cos( \theta ) +e_{j}\Vert v\Vert \sin( \theta )$?

Question

I know two non-colinear N-dimensional vectors always describe a plane. Intuitively, a vector $u$ should always allow to be written as a rotation from a reference vector in an arbitrary 2-dimensional orthonormal basis, one basis vector being colinear with the reference vector.

So, can a vector $u$ be always written as $e_{i}\Vert v\Vert \cos( \theta ) +e_{j}\Vert v\Vert \sin( \theta )$ with $e_i$ and $e_j$ being orthonormal basis vectors and $\theta$ the angle between $u$ and $e_i$?

Answer 1

Almost:\begin{align}v&=\langle v,e_1\rangle e_1+\langle v,e_2\rangle e_2\\&=\|v\|\left(\left\langle\frac v{\|v\|},e_1\right\rangle e_1+\left\langle\frac v{\|v\|},e_2\right\rangle e_2\right)\\&=\|v\|\bigl(\cos(\theta)e_1\pm\sin(\theta)e_2\bigr).\end{align}That $\pm$ sign may well be negative.