geometrymatricesvectors
I need to rotate a vector using transformation matrix. For example: I heave vector Z (0, 0, 1). I'm rotating it by 100 deg around Z-Axis. Result will be the same as input. How to compute the angle which vector was rotated around own axis?
I think I need more support vectors (like I painted on pic above). Of course I'm not rotatig around 3 main axes. Rotation can be any in 3D.
If $R_x$ rotates around the $x$-axis, and $R_y$ rotates around the $y$-axis, and you want to rotate first around $x$, and then around $y$, simply apply $R_y R_x$ to your vector, let's call it $v$.
This is because $R_x v$ rotates $v$ around the $x$-axis, then $R_y(R_x v)$ rotates $R_x v$ around the $y$-axis.
We refer to the figure below.
Let $V_1$ and $V_2 $ be the normalized vectors $A$ and $B$ (unit associated vectors). It suffices to know how to transform $V_1$ into $V_2$, with resizing at the begining and at the end. So we are going to work on the unit sphere.
A solution to your problem is by the following succession of rotations $R_x$ followed by $R_z$ followed by $R_y$:
$R_x$ brings $V_1$ onto a vector $R_x(V_1)$ of the equatorial (xOy) plane.
$(R_y)^{-1}$ (take care of the $-1$) brings $V_2$ onto another vector $(R_y)^{-1}(V_2)$ of the equatorial (xOy) plane.
$R_z$ brings $R_x(V_1)$ onto $(R_y)^{-1}(V_2)$.
The consequence is that $R_z(R_x(V_1))=(R_y)^{-1}V_2$, meaning that
$$R_y(R_z(R_x(V_1)))=V_2$$
as desired.
I do not enter into computational details: the angles of rotations $R_x,R_y,R_z$ are easily obtained. But we can have a discussion about it.
Best Answer
Check out this Wikipedia article on the rotation of vectors in $\mathbb{R}^3$. It describes the transformation matrix used for the rotation of a vector about an arbitrary unit vector by an arbitrary angle.
Regarding the computation of the rotation angle, we consider the rotation of an arbitrary vector $\vec{A}$ about some unit vector (the rotation axis) by an angle of $\theta$ to get $\vec{B}$. The dot product of the two gives $\vec{A}\cdot\vec{B} = |\vec{A}||\vec{B}|\cos{\theta}$, or more explicitly,
$$\cos{\theta} = \frac{\vec{A}\cdot\vec{B}}{|\vec{A}||\vec{B}|}$$ This allows you to compute the rotation angle ($\theta$) from the vectors before and after rotation.
As for the computation of the rotation axis, we simply take and normalize the cross product of $\vec{A}$ with $\vec{B}$. I hope you can see why.