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{\bf Question}

Let $$A = \left( \begin{array}{ccc}  \ds \frac{1}{\sqrt3}  & \ds
\frac{1}{\sqrt2} & \ds -\frac{1}{\sqrt6} \\ \ds \frac{1}{\sqrt3} &
\ds -\frac{1}{\sqrt2} & \ds -\frac{1}{\sqrt6} \\ \ds
\frac{-1}{\sqrt3} & 0 & \ds -\frac{2}{\sqrt6} \end{array}
\right).$$ Verify that $A$ is orthogonal.  Suppose co-ordinates
are related by ${\bf x} = A{\bf X}$.  Find the ${\bf X}$ equations
of the images of the $x_1, \, x_2$ and $x_3$ axes and verify that
the images are mutually orthogonal.

\vspace{.25in}

{\bf Answer}

Verify $A$ orthogonal $A^TA=I$

$\left( \begin{array}{c} X_1 \\ X_2 \\ X_3 \end{array} \right) =
\left( \begin{array}{ccc} \frac{1}{\sqrt3} & \frac{1}{\sqrt3} &
-\frac{1}{\sqrt3} \\ \frac{1}{\sqrt2} & -\frac{1}{\sqrt3} & 0 \\
-\frac{1}{\sqrt6} & -\frac{1}{\sqrt6} & -\frac{2}{\sqrt6}
\end{array} \right) \left( \begin{array}{c} x_1 \\ x_2 \\x_3
\end{array} \right)$

${}$

So the images are:-

$\left. \begin{array}{l} x_1 {\rm \ axis\ } \left(
\frac{1}{\sqrt3}x_1 , \frac{1}{\sqrt 2}x_1, -\frac{1}{\sqrt6}x_1
\right) \\ \\ x_2 {\rm \ axis\ } \left( \frac{1}{\sqrt3}x_2 ,
-\frac{1}{\sqrt 2}x_2, -\frac{1}{\sqrt6}x_1 \right) \\ \\ x_3 {\rm
\ axis\ } \left( -\frac{1}{\sqrt3}x_3 , 0, -\frac{2}{\sqrt6}x_3
\right) \end{array} \right\}$ Mutually orthogonal.

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