I know the three components of a vector, say $u_{x}$, $u_{y}$ and $u_{z}$ about a coordiante system prior to rotation. Their resultant gives me the vector $u_{res}$. The direction of this resultant vector is the direction of my new X axis, i.e. the X axis is rotated to align it along the resultant vector.
How do I find the components of any vector in this new coordinate system?
Not really an answer, but I wanted to include an ugly illustration.
As it is, your question is probably underspecified, since there is an infinity of possible 3D rotations that can map the positive $x$-axis onto your (resultant) vector $\mathbf u$. If you only care about mapping the (old) $x$-axis onto a new direction, these infinite possibilities don't matter. But because you actually need to also figure out what your new $y$ and $z$-axes are, these infinite possibilities become a problem.
Say your original axes are supported by the vectors $\mathbf i$, $\mathbf j$ and $\mathbf k$ (in red). One possible rotation that maps $\mathbf i$ onto $\mathbf u$ is represented by the solid black arc, and will map these vectors onto $\mathbf i'$, $\mathbf j'$ and $\mathbf k'$ (blue).
But if you pick another rotation that still maps $\mathbf i$ to $\mathbf u$, you will end up with completely different vectors $\mathbf j'$ and $\mathbf k'$ :
And different basis vectors imply different coordinates.
On a side note, if someone knows a good software to make simple 3D illustrations, I'd like some recommendations.