Question

(a) Prove that the two-dimensional rotation matrix (Eq.1.29) preserves dot products.
(That is, show that$\overline{{\mathbf{A}}_{\mathbf{y}}{\mathbf{B}}_{\mathbf{y}}}\mathbf{+}\overline{{\mathbf{A}}_{\mathbf{z}}{\mathbf{B}}_{\mathbf{z}}}\mathbf{=}{\mathit{A}}_{\mathbf{y}}{\mathit{B}}_{\mathbf{y}}\mathbf{+}{\mathit{A}}_{\mathbf{z}}{\mathit{B}}_{\mathbf{z}}$.)
(b) What constraints must the elements (Rij) of the three-dimensional rotation matrix
(Eq.1.30) satisfy, in order to preserve the length of A (for all vectors$\overrightarrow{\mathbf{A}}$ )?

Short answer

(a) It is proved that two dimensional rotation matrix preserves dot product

(b) The constraints in order to preserve the length of A are:

$$\begin{gathered} 2\left(R_{xy}{R}_{xz+}{R}_{yy}{R}_{yz}+R_{zy}{R}_{zz}\right)=0 \\ 2\left(R_{xz}{R}_{xx+}{R}_{yz}{R}_{yx}+R_{zz}{R}_{zx}\right)=0 \\ 2\left(R_{xy}{R}_{xz+}{R}_{yy}{R}_{yz}+R_{zy}{R}_{zz}\right)=0 \\ 2\left(R_{xx}{R}_{xy+}{R}_{xy}{R}_{yy}+R_{zx}{R}_{zy}\right)=0 \end{gathered}$$

And

$$\begin{gathered} \left({R_{xx}}^2+{R_{yx}}^2+{R_{zx}}^2\right)=0 \\ \left({R_{xy}}^2+{R_{yy}}^2+{R_{zy}}^2\right)=0 \\ \left({R_{xz}}^2+{R_{yz}}^2+{R_{zz}}^2\right)=0 \end{gathered}$$

Step-by-step solution

Explain the concept and write the 2-D matix.

Thescalars are invariant under rotations. Mass m is always the same underrotations.

Also, dot product is a scalar quantity. So, it should be invariant under rotation. Vectors transformation of rotation around xaxisisgivenby,

$$\begin{gathered} \overrightarrow{A_y}=\cos \varphi A_y+\sin \varphi A_z \\ \overrightarrow{A_z}=-\sin \varphi A_y+\cos \varphi A_z \end{gathered}$$

Similarly, find $\overrightarrow{B_y}$and$\overrightarrow{B_z}$

$$\begin{gathered} \overrightarrow{B_y}=\cos \varphi B_y+\sin \varphi B_z \\ \overrightarrow{B_z}=-\sin \varphi B_y+\cos \varphi B_z \end{gathered}$$

The dot product of two dimensional rotational vector

Hence, the two-dimensional rotation matrix preserves dot products.

Find the constraint on rotation matrix

Since the coordinate system is rotated around an arbitrary axis, Origin is always at the same point in both the coordinate system. So length of the vector should be invariant. Under rotation.

Forrotationaboutanarbitraryaxisinthreedimensions, thetransformationlawis,

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

Compare the length of the system in the two coordinated system as follows:

$$\begin{gathered} \sqrt{{\overline{A_x}}^2+{\overline{A_y}}^2+{\overline{A_z}}^2}=\sqrt{{A_x}^2+{A_y}^2+{A_z}^2} \\ {\overline{A_x}}^2+{\overline{A_y}}^2+{\overline{A_z}}^2={A_x}^2+{A_y}^2+{A_z}^2 \end{gathered}$$

Let us consider

localid="1657357736690" ${\overline{A_x}}^2={R_{xx}}^2{A_x}^2+{R_{xy}}^2{A_z}^2+2R_{xx}{R}_{xy}{A}_x{A}_y+2R_{xy}{R}_{xz}{A}_y{A}_z+2R_{xz}{R}_{xx}{A}_z{A}_x$

Similarly, we can write

$$\begin{gathered} {\overline{A_y}}^2={R_{yy}}^2{A_x}^2+{R_{xy}}^2{A_y}^2+{R_{xz}}^2{A_z}^2+2R_{xx}{R}_{xy}{A}_x{A}_y+2R_{xy}{R}_{xz}{A}_y{A}_z+2R_{xz}{R}_{xx}{A}_z{A}_x \\ {\overline{A_z}}^2={R_{zz}}^2{A_z}^2+{R_{zy}}^2{A_y}^2+{R_{zx}}^2{A_x}^2+2R_{zz}{R}_{zy}{A}_z{A}_y+2R_{zy}{R}_{zx}{A}_y{A}_z+2R_{zx}{R}_{zz}{A}_z{A}_y \end{gathered}$$

So,

$$\begin{gathered} {\overline{A_x}}^2+{\overline{A_y}}^2+{\overline{A_z}}^2=\left({R_{xx}}^2+{R_{yx}}^2+{R_{zx}}^2\right){A_x}^2+\left({R_{xy}}^2+{R_{yy}}^2+{R_{zy}}^2\right){A_y}^2\left({R_{xz}}^2+{R_{yz}}^2+{R_{zz}}^2\right){A_z}^2 \\ +\left(R_{xx}{R}_{xy+}{R}_{xy}{R}_{yy}+R_{zx}{R}_{zy}\right)A_y{A}_x+2\left(R_{xy}{R}_{xz+}{R}_{yy}{R}_{yz}+R_{zy}{R}_{zz}\right)A_y{A}_z \\ +2\left(R_{xy}{R}_{xz+}{R}_{yy}{R}_{yz}+R_{zy}{R}_{zz}\right)A_y{A}_z+2\left(R_{xz}{R}_{xx+}{R}_{yz}{R}_{yx}+R_{zz}{R}_{zx}\right)A_z{A}_x \end{gathered}$$

On comparing this equation with equation (1), we get

$$\begin{gathered} 2\left(R_{xy}{R}_{xz+}{R}_{yy}{R}_{yz}+R_{zy}{R}_{zz}\right)=0 \\ 2\left(R_{xz}{R}_{xx+}{R}_{yz}{R}_{yx}+R_{zz}{R}_{zx}\right)=0 \\ 2\left(R_{xy}{R}_{xz+}{R}_{yy}{R}_{yz}+R_{zy}{R}_{zz}\right)=0 \\ 2\left(R_{xx}{R}_{xy+}{R}_{xy}{R}_{yy}+R_{zx}{R}_{zy}\right)=0 \end{gathered}$$

And also, we obtain,

$$\begin{gathered} \left({R_{xx}}^2+{R_{yx}}^2+{R_{zx}}^2\right)=0 \\ \left({R_{xy}}^2+{R_{yy}}^2+{R_{zy}}^2\right)=0 \\ \left({R_{xz}}^2+{R_{yz}}^2+{R_{zz}}^2\right)=0 \end{gathered}$$

Thus these results are the constraints of rotation matrix.