How can I calculate the components of a 3D vector after it has been rotated I have a 3D vector, with the knowledge of its current 3 components and therefore can calculate its magnitude and direction cosines.
They may change but for the sake of the question they can be <1,3,-2>. The origin is at 0,0,0.
By rotating the vector by an angle, how can I find the components after the vector has been rotated by the angle theta. All axes may need to be rotated.
Note: The vector may be rotated in more then one angle at a time (angles α & γ might be rotated together and not separately)
 A: Rotations around coordinate axes can be done using matrices. Let $\theta$ be the angle.


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*Clockwise rotation around $x$-axis:
$$\begin{bmatrix} x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix}$$

*Clockwise rotation around $y$-axis:
$$\begin{bmatrix} x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} \cos\theta & 0 & -\sin\theta \\ 0 & 1 & 0 \\ \sin\theta & 0 & \cos\theta \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix}$$

*Clockwise rotation around $z$-axis:
$$\begin{bmatrix} x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix}$$
Of course, you can rotate around several axes. In that case, you just need to use the corresponding matrices kkeping the order. For instance, if you want to rotate $\alpha$ around $y$-axis first and then $\beta$ around $x$-axis:
$$\begin{bmatrix} x'' \\ y'' \\ z'' \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\beta & -\sin\beta \\ 0 & \sin\beta & \cos\beta \end{bmatrix}\begin{bmatrix} x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\beta & -\sin\beta \\ 0 & \sin\beta & \cos\beta \end{bmatrix}\begin{bmatrix} \cos\alpha & 0 & -\sin\alpha \\ 0 & 1 & 0 \\ \sin\alpha & 0 & \cos\alpha \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix}$$
