Rotation matrices are used to rotate a vector
into a new direction.

In transforming vectors in three-dimensional space, rotation
matrices are often encountered. Rotation matrices are used in two
senses: they can be used to rotate a vector into a new position or
they can be used to rotate a coordinate basis (or coordinate system)
into a new one. In this case, the vector is left alone but its components
in the new basis will be different from those in the original basis.
In Euclidean space, there are three basic rotations: one each around
the x, y and z axes. Each rotation is specified by an angle of rotation.
The rotation angle is defined to be positive for a rotation that is
counterclockwise when viewed by an observer looking along the rotation
axis towards the origin. Any arbitrary rotation can be composed of
a combination of these three *(Euler's rotation theorem)*.
For example, one can rotated a vector using a sequence of three rotations: $${v}^{\prime}=Av={R}_{z}(\gamma ){R}_{y}(\beta ){R}_{x}(\alpha )v$$.

The rotation matrices that rotate a vector around the x, y,
and z-axes are given by:

Counterclockwise rotation around x-axis

$${R}_{x}(\alpha )=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\alpha & -\mathrm{sin}\alpha \\ 0& \mathrm{sin}\alpha & \mathrm{cos}\alpha \end{array}\right]$$

Counterclockwise rotation around y-axis

$${R}_{y}(\beta )=\left[\begin{array}{ccc}\mathrm{cos}\beta & 0& \mathrm{sin}\beta \\ 0& 1& 0\\ -\mathrm{sin}\beta & 0& \mathrm{cos}\beta \end{array}\right]$$

Counterclockwise rotation around z-axis

$${R}_{z}(\gamma )=\left[\begin{array}{ccc}\mathrm{cos}\gamma & -\mathrm{sin}\gamma & 0\\ \mathrm{sin}\gamma & \mathrm{cos}\gamma & 0\\ 0& 0& 1\end{array}\right]$$

The following three figures show what positive rotations look
like for each rotation axis:

For any rotation, there is an inverse rotation satisfying $${A}^{-1}A=1$$.
For example, the inverse of the x-axis rotation matrix is obtained
by changing the sign of the angle:

$${R}_{x}^{-1}(\alpha )={R}_{x}(-\alpha )=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\alpha & \mathrm{sin}\alpha \\ 0& -\mathrm{sin}\alpha & \mathrm{cos}\alpha \end{array}\right]={R}_{x}^{\prime}(\alpha )$$

This example illustrates
a basic property: the inverse rotation matrix equals the transpose
of the original. Rotation matrices satisfy *A'A = 1*,
and consequently *det(A) = 1*. Under rotations, vector
lengths are preserved as well as the angles between vectors.

We can think of rotations in another way. Consider the original
set of basis vectors, $$i,j,k$$,
and rotate them all using the rotation matrix *A*.
This produces a new set of basis vectors $${i}^{\prime},j{,}^{\prime}{k}^{\prime}$$ related
to the original by:

$$\begin{array}{ll}{i}^{\prime}\hfill & =Ai\hfill \\ {j}^{\prime}\hfill & =Aj\hfill \\ {k}^{\prime}\hfill & =Ak\hfill \end{array}$$

The new basis vectors
can be written as linear combinations of the old ones and involve
the transpose:

$$\left[\begin{array}{c}{i}^{\prime}\\ {j}^{\prime}\\ {k}^{\prime}\end{array}\right]={A}^{\prime}\left[\begin{array}{c}i\\ j\\ k\end{array}\right]$$

Now any vector can be written as a linear combination of either
set of basis vectors:

$$v={v}_{x}i+{v}_{y}j+{v}_{z}k={{v}^{\prime}}_{x}{i}^{\prime}+{{v}^{\prime}}_{y}{j}^{\prime}+{{v}^{\prime}}_{z}{k}^{\prime}$$

Using some algebraic
manipulation, one can derive the transformation of components for
a fixed vector when the basis (or coordinate system) rotates

$$\left[\begin{array}{c}{{v}^{\prime}}_{x}\\ {{v}^{\prime}}_{y}\\ {{v}^{\prime}}_{z}\end{array}\right]={A}^{-1}\left[\begin{array}{c}{v}_{x}\\ {v}_{y}\\ {v}_{z}\end{array}\right]={A}^{\prime}\left[\begin{array}{c}{v}_{x}\\ {v}_{y}\\ {v}_{z}\end{array}\right]$$

Thus the change in
components of a vector when the coordinate system rotates involves
the transpose of the rotation matrix. The next figure illustrates
how a vector stays fixed as the coordinate system rotates around the
x-axis. The figure after shows how this can be interpreted as a rotation *of
the vector * in the opposite direction.