Can we do the same type of operation in vectors like we do in complex numbers and vice-versa? We know that e^itheta in complex numbers is very special because it makes the complex numbers to rotate some angle about origin . Is there a similar thing present in vectors too which rotate a vector by some angle? And what about the cross product in vectors , can we be able to find a complex number which is perpendicular to plane formed by some two complex numbers and the origin?
 A: Operation  $z \to e^{i \alpha}z$ has a close equivalent in 3D.
Indeed, the rotation with angle $\theta$ about unit vector $V_0:= \begin{pmatrix}a\\b\\c \end{pmatrix}$ is
$$R_{\theta,V_0}=\exp(\theta [V_0]_{\times})\tag{1}$$
where $\exp(M)=I+M+\frac12 M^2+ \frac{1}{3!} M^3+\cdots$ is the matrix exponential and $[V_0]_X$ is the antisymmetric matrix defined by
$$[V_0]_{\times}=\begin{pmatrix}0&-c&b\\c&0&-a\\-b&a&0\end{pmatrix}\tag{2}$$
Important remark: One could define $[V_0]_{\times}$ by the following formula :
$$[V_0]_{\times}V=V_0 \times V\tag{2}$$
In detail:
$$\begin{pmatrix}0&-c&b\\c&0&-a\\-b&a&0\end{pmatrix}\begin{pmatrix}x\\y\\z \end{pmatrix}=\begin{pmatrix}-cy+bz\\ \ \ cx-az\\-bx+ay \end{pmatrix}$$
A: The quaternions may be used in place of complex numbers to describe three dimensional rotations. See here.
Let $\vec{u}$ be a unit vector (the rotation axis) and consider the quaternion $q = \cos \frac{\alpha}{2} + \vec{u} \sin \frac{\alpha}{2}$. Then
$$\vec{v'} = q \vec{v} q^{-1} = \left( \cos \frac{\alpha}{2} + \vec{u} \sin \frac{\alpha}{2} \right) \, \vec{v} \, \left( \cos \frac{\alpha}{2} - \vec{u} \sin \frac{\alpha}{2} \right)$$
yields the vector $\vec{v}$ rotated by an angle $\alpha$ around the axis $\vec{u}$.
