Matrix of axes is a rotation matrix

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Matrix with the axes of orthonormal coordinates is a rotation matrix. This matrix can convert from a certain coordinates to another certain coordinates.

For example, let's transform a point in x,y,z axes 3D orthonormal space to a point in a,b,c axes 3D orthonormal space. Through this conversion, we assume that the x axis becomes a axis, y axis becomes b axis, and z axis becomes c axis.

$$\begin{array}{c}a={\left(\begin{array}{ccc}{a}_x& a_y& a_z\end{array}\right)}^T\\ b={\left(\begin{array}{ccc}{b}_x& b_y& b_z\end{array}\right)}^T\\ c={\left(\begin{array}{ccc}{c}_x& c_y& c_z\end{array}\right)}^T\end{array}$$

This transform matrix is expressed as follows.

$$\left(\begin{array}{ccc}{a}_x& b_x& c_x\\ a_y& b_y& c_y\\ a_z& b_z& c_z\end{array}\right)$$

Check.

$$x={\left(\begin{array}{ccc}1& 0& 0\end{array}\right)}^T$$

$$\left(\begin{array}{ccc}{a}_x& b_x& c_x\\ a_y& b_y& c_y\\ a_z& b_z& c_z\end{array}\right)\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)=\left(\begin{array}{c}{a}_x\\ a_y\\ a_z\end{array}\right)$$

Muliplying this matrix to x becomes a.

$$y={\left(\begin{array}{ccc}0& 1& 0\end{array}\right)}^T$$

$$\left(\begin{array}{ccc}{a}_x& b_x& c_x\\ a_y& b_y& c_y\\ a_z& b_z& c_z\end{array}\right)\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)=\left(\begin{array}{c}{b}_x\\ b_y\\ b_z\end{array}\right)$$

Muliplying this matrix to y becomes b.

$$z={\left(\begin{array}{ccc}0& 0& 1\end{array}\right)}^T$$

$$\left(\begin{array}{ccc}{a}_x& b_x& c_x\\ a_y& b_y& c_y\\ a_z& b_z& c_z\end{array}\right)\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)=\left(\begin{array}{c}{c}_x\\ c_y\\ c_z\end{array}\right)$$

Muliplying this matrix to z becomes c.

This transform matrix converts x,y,z space to a,b,c space.

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