Q9P - Introduction to Electrodynamics

Question

Find the transformation matrix R that describes a rotation by 120° about an axis from the origin through the point (1, 1, 1). The rotation is clockwise as you look down the axis toward the origin.

Step-by-Step Solution

Answer

The rotation matrix is obtained as :

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

Step 1: Explain the concept and draw the cube using given information

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.

Step 2: Draw the original position of the axes

The original position of the axes are drawn as,

                                   

Step 3: Draw the new position of the axes

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

                                              

Step 4: Describe the new position of the axes

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.

Step 5: Write the transformation matrix

The transformation matrix takes the following form:

 

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

 

Due to transformation, 

 

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

Step 6: Compare LHS and RHS of equation (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 :

 

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