Chapter 3.9, Problem 24P

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