Answer
${\bf{i}} \times {\bf{j}} = {\bf{k}}$, ${\ \ }$ ${\bf{j}} \times {\bf{k}} = {\bf{i}}$, ${\ \ }$ ${\bf{k}} \times {\bf{i}} = {\bf{j}}$
${\bf{i}} \times {\bf{i}} = {\bf{j}} \times {\bf{j}} = {\bf{k}} \times {\bf{k}} = {\bf{0}}$
Work Step by Step
From Eq. (5) we have
${\bf{i}} \times {\bf{j}} = {\bf{k}}$, ${\ \ }$ ${\bf{j}} \times {\bf{k}} = {\bf{i}}$, ${\ \ }$ ${\bf{k}} \times {\bf{i}} = {\bf{j}}$
${\bf{i}} \times {\bf{i}} = {\bf{j}} \times {\bf{j}} = {\bf{k}} \times {\bf{k}} = {\bf{0}}$
We verify this by using the components of ${\bf{i}}$, ${\bf{j}}$ and ${\bf{k}}$:
${\bf{i}} = \left( {1,0,0} \right)$, ${\ \ }$ ${\bf{j}} = \left( {0,1,0} \right)$, ${\ \ }$ ${\bf{k}} = \left( {0,0,1} \right)$
Using the formula for the cross product, we obtain
${\bf{i}} \times {\bf{j}} = \left| {\begin{array}{*{20}{c}}
{\bf{i}}&{\bf{j}}&{\bf{k}}\\
1&0&0\\
0&1&0
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
0&0\\
1&0
\end{array}} \right|{\bf{i}} - \left| {\begin{array}{*{20}{c}}
1&0\\
0&0
\end{array}} \right|{\bf{j}} + \left| {\begin{array}{*{20}{c}}
1&0\\
0&1
\end{array}} \right|{\bf{k}}$
${\bf{i}} \times {\bf{j}} = {\bf{k}}$
${\bf{j}} \times {\bf{k}} = \left| {\begin{array}{*{20}{c}}
{\bf{i}}&{\bf{j}}&{\bf{k}}\\
0&1&0\\
0&0&1
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
1&0\\
0&1
\end{array}} \right|{\bf{i}} - \left| {\begin{array}{*{20}{c}}
0&0\\
0&1
\end{array}} \right|{\bf{j}} + \left| {\begin{array}{*{20}{c}}
0&1\\
0&0
\end{array}} \right|{\bf{k}}$
${\bf{j}} \times {\bf{k}} = {\bf{i}}$
${\bf{k}} \times {\bf{i}} = \left| {\begin{array}{*{20}{c}}
{\bf{i}}&{\bf{j}}&{\bf{k}}\\
0&0&1\\
1&0&0
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
0&1\\
0&0
\end{array}} \right|{\bf{i}} - \left| {\begin{array}{*{20}{c}}
0&1\\
1&0
\end{array}} \right|{\bf{j}} + \left| {\begin{array}{*{20}{c}}
0&0\\
1&0
\end{array}} \right|{\bf{k}}$
${\bf{k}} \times {\bf{i}} = {\bf{j}}$
${\bf{i}} \times {\bf{i}} = \left| {\begin{array}{*{20}{c}}
{\bf{i}}&{\bf{j}}&{\bf{k}}\\
1&0&0\\
1&0&0
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
0&0\\
0&0
\end{array}} \right|{\bf{i}} - \left| {\begin{array}{*{20}{c}}
1&0\\
1&0
\end{array}} \right|{\bf{j}} + \left| {\begin{array}{*{20}{c}}
1&0\\
1&0
\end{array}} \right|{\bf{k}}$
${\bf{i}} \times {\bf{i}} = {\bf{0}}$
${\bf{j}} \times {\bf{j}} = \left| {\begin{array}{*{20}{c}}
{\bf{i}}&{\bf{j}}&{\bf{k}}\\
0&1&0\\
0&1&0
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
1&0\\
1&0
\end{array}} \right|{\bf{i}} - \left| {\begin{array}{*{20}{c}}
0&0\\
0&0
\end{array}} \right|{\bf{j}} + \left| {\begin{array}{*{20}{c}}
0&1\\
0&1
\end{array}} \right|{\bf{k}}$
${\bf{j}} \times {\bf{j}} = {\bf{0}}$
${\bf{k}} \times {\bf{k}} = \left| {\begin{array}{*{20}{c}}
{\bf{i}}&{\bf{j}}&{\bf{k}}\\
0&0&1\\
0&0&1
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
0&1\\
0&1
\end{array}} \right|{\bf{i}} - \left| {\begin{array}{*{20}{c}}
0&1\\
0&1
\end{array}} \right|{\bf{j}} + \left| {\begin{array}{*{20}{c}}
0&0\\
0&0
\end{array}} \right|{\bf{k}}$
${\bf{k}} \times {\bf{k}} = {\bf{0}}$
