<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
\parindent=0pt
\begin{document}


{\bf Question}

A right-handed rectangular co-ordinate system is rotated through
an angle of $120^\circ$ about the line $x = y = z,$  Find the
matrix $A$ of the transformation and show that $\det A = +1$. What
is $A^3$?

\vspace{.25in}

{\bf Answer}

If we perform this rotation successively 3 times we get back to
where we started from.  So $A^3 = I$.

The rotation permutes the axis $x_1x_2x_3 \rightarrow x_3x_1x_2$
So $$\left( \begin{array}{c} x_3 \\ x_1 \\ x_2 \end{array} \right)
= \left( \begin{array}{ccc} 0 &amp; 0 &amp; 1 \\ 1 &amp;  0 &amp; 0 \\ 0 &amp; 1 &amp; 0
\end{array} \right) \left( \begin{array}{c} x_1 \\ x_2 \\x_3
\end{array} \right)$$ $\det A = 1 \hspace{.2in} A^2 = \left(
\begin{array}{ccc} 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1 \\ 1 &amp; 0 &amp; 0 \end{array}
\right)$

If the rotation is the other way $x_1x_2x_3 \rightarrow x_2
x_3x_1$ then $A = \left( \begin{array}{ccc} 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1
\\ 1 &amp; 0 &amp; 0\end{array} \right)$


\end{document}
</pre></body></html>