Question

Show that an orthogonal transformation preserves the length of vectors. Hint: Ifis the column matrix of vector[see (6.10)], write out${\mathbf{\text{r}}}^{\mathbf{\text{T}}}\mathbf{\text{r}}$to show that it is the square of the length of r. Similarly${\mathbf{\text{R}}}^{\mathbf{\text{T}}}\mathbf{\text{R}}\mathbf{=}\mathbf{|}\mathbf{R}{\mathbf{|}}^{\mathbf{2}}$and you want to show that$\mathbf{|}\mathbf{R}{\mathbf{|}}^{\mathbf{2}}\mathbf{=}\mathbf{|}\mathbf{r}{\mathbf{|}}^{\mathbf{2}}$, that is ${\mathit{R}}^{\mathbf{\text{T}}}\mathit{R}\mathbf{=}{\mathit{r}}^{\mathbf{\text{T}}}\mathbf{\text{r}}$,if$\mathbf{\text{R}}\mathbf{=}\mathbf{\text{Mr}}$and Mis orthogonal. Use (9.11).

Short answer

The length of the vectors is preserved by the orthogonal transformation that is, $|R{|}^2=|r{|}^2$.

Step-by-step solution

Given information.

$$\begin{gathered} R^{T}R=|R{|}^2 \\ {R}^{T}R=r^{T}r\text{if}R=Mr \end{gathered}$$

Orthogonal transformation

An orthogonal transformation is a linear transformation that keeps the inner product symmetric. An orthogonal transformation, in particular, preserves vector lengths and angles between vectors.

Show that the length of the vectors is preserved by the orthogonal transformation that is, |R|2=|r|2.

The lengths of the vectors are preserved by the orthogonal transformation.

$$\begin{gathered} \overrightarrow{r}=r=\left(\begin{array}{c}x\\ y\\ z\end{array}\right) \\ \overrightarrow{r}=r=\left(\begin{array}{ccc}x& y& z\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right) \\ =x^2+y^2+z^2 \\ =|r{|}^2 \end{gathered}$$

Assume the value of the matrix R as shown below.

$$\begin{gathered} R=\left(\begin{array}{c}X\\ Y\\ Z\end{array}\right) \\ {R}^{T}R=\left(\begin{array}{ccc}& Y& Z\end{array}\right)\left(\begin{array}{c}X\\ Y\\ Z\end{array}\right) \\ =X^2+Y^2+Z^2 \\ =|R{|}^2 \end{gathered}$$

Here,

$$\begin{gathered} R=Mr \\ R^T=r^T{M}^T \\ {R}^{T}R=r^T{M}^{T}Mr \\ R^{T}R=r^{T}r \end{gathered}$$

Solve further and get,

$|R{|}^2=|r{|}^2$

Hence, proved.