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Definition of Cross product
\begin{eqnarray}
\bm{c} = \bm{a} \times \bm{b}= ( |\bm{a}||\bm{b}| sin \theta)\bm{e}
\end{eqnarray}
Unit vector for $x,y,z$ axis : $\bm{i},\bm{j},\bm{k}$
\begin{eqnarray}
\bm{i} \times \bm{i} = 0,\ \bm{j} \times \bm{j} = 0,\ \bm{k} \times \bm{k} = 0\\
\bm{i} \times \bm{j} = \bm{k},\ \bm{j} \times \bm{k} = \bm{i},\ \bm{k} \times \bm{i} = \bm{j}\\
\bm{j} \times \bm{i} = -\bm{k},\ \bm{k} \times \bm{j} = -\bm{i},\ \bm{i} \times \bm{k} = -\bm{j}
\end{eqnarray}
Elements of cross product
\begin{eqnarray}
\lefteqn{\bm{a} \times \bm{b} }\nonumber \\
&= &(a_x\bm{i} + a_y\bm{j} + a_z\bm{k}) \times (b_x\bm{i} + b_y\bm{j} + a_z\bm{k}) \nonumber \\
&=&(a_y a_z - a_z b_y)\bm{i} + (a_z b_x - a_x b_z)\bm{j} + (a_x b_y - a_y b_x)\bm{k}
\end{eqnarray}
Cross product using vector determinant
\begin{eqnarray}
\bm{a} \times \bm{b} =\left| \begin{array}{ccc}
\bm{i} & \bm{j} & \bm{k} \\
a_x & a_y & a_z \\
b_x & b_y & b_z
\end{array} \right|
\end{eqnarray}
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