91Ó°ÊÓ

During rotation, the axes of a coordinate system are rotated counter clockwise through a given angle. The transformed axes in the rotated coordinate system are now fed to output of the former matrix system to determine the rotation matrix.

It is given that angle of rotation is $120°$, the point is $M=\left(1,1,1\right)$and the direction of rotation is clockwise.

The original position of the axes are drawn as,

The new position of the axes, after clockwise rotation of $120°$are drawn as,

Due to new position of the axes, after clockwise rotation of $120°$, the z-axis is shifted to y- axis, x-axis is shifted to z- axis, and y-axis is shifted to x axis.

The transformation matrix takes the following form:

$\left[\begin{array}{c}\stackrel{¯}{A_x}\\ \stackrel{¯}{A_y}\\ \stackrel{¯}{A_z}\end{array}\right]=\left[\begin{array}{ccc}{R}_{xx}& R_{xy}& R_{xz}\\ R_{yx}& R_{yy}& R_{yz}\\ R_{zx}& R_{zy}& Rz\end{array}\right]\left[\begin{array}{c}{A}_x\\ A_y\\ A_z\end{array}\right]$ …….. (1)

Due to transformation,

$\left[\begin{array}{c}\stackrel{¯}{A_x}\\ \stackrel{¯}{A_y}\\ \stackrel{¯}{A_z}\end{array}\right]=\left[\begin{array}{c}{A}_z\\ A_x\\ A_y\end{array}\right]$ …… (2)

Combine equatiions (1) and (2), as

$\left[\begin{array}{c}\begin{array}{c}{A}_z\\ A_x\\ A_y\end{array}\end{array}\right]=\left[\begin{array}{ccc}{R}_{xx}& R_{xy}& R_{xz}\\ R_{yx}& R_{yy}& R_{yz}\\ R_{zx}& R_{zy}& Rz\end{array}\right]\left[\begin{array}{c}{A}_x\\ A_y\\ A_z\end{array}\right]$

The matrix can be simplified as,

$\left.\begin{array}{c}{A}_z=R_{xx}{A}_x+R_{xy}{A}_y+R_{xz}{R}_z\\ A_x=R_{yx}{A}_x+R_{yy}{A}_y+R_{yz}{R}_z\\ A_y=R_{zx}{A}_x+R_{zy}{A}_y+R_{zz}{R}_z\end{array}\right\}$ …… (3)

On comparing LHS and RHS of equation (3), we get,

$$\begin{gathered} R_{xx}=0,R_{xy}=0,R_{xz}=1 \\ {R}_{yx}=1,R_{yy}=0,R_{yz}=0 \\ {R}_{zx}=0,R_{zy}=1,R_{zz}=0 \end{gathered}$$

Thus, the rotation matrix is obtained as :

$\left[\begin{array}{ccc}{R}_{xx}& R_{xy}& R_{xz}\\ R_{yx}& R_{yy}& R_{yz}\\ R_{zx}& R_{zy}& R_{zz}\end{array}\right]=\left[\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right]$