Properties of cross Product:
1) If \(
\hat{i},\hat{j},\hat{k}
\) are unit vectors then
\(
\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = \overrightarrow 0
\) , \(
\hat{i} \times \hat{j} = \hat{k},\hat{j} \times \hat{k} = \hat{i},\hat{k} \times \hat{i} = \hat{j}
\) , \(
\hat{j} \times \hat{i} = - \hat{k},\hat{k} \times \hat{j} = - \hat{i},\,\,\hat{i} \times \hat{k} = - \hat{j}
\)
2) If \(
\overrightarrow A = A_x \hat i + A_y \hat j + A_z \hat k
\) and \(
\overrightarrow B = B_x \hat i + B_y \hat j + B_z \hat k
\)
then \(
\overrightarrow A \times \overrightarrow B = \left| {\begin{array}{*{20}c}
{\hat i} & {\hat j} & {\hat k} \\
{A_x } & {A_y } & {A_z } \\
{B_x } & {B_y } & {B_z } \\
\end{array} } \right|
\) \(
= (A_y B_z - A_z B_y )\hat i - (A_x B_z - A_z B_x )\hat j + (A_x B_y - A_y B_x )\hat k
\)
3) Unit vector normal to both \(\vec A\) and \(\vec B\) is \(
\hat{n} = \frac{{\overrightarrow A \times \overrightarrow B }}
{{\left| {\overrightarrow A \times \overrightarrow B } \right|}}
\).
4) If two vectors are parallel (\(\theta=0\)) or anti parallel (\(\theta=180^0\) ) then \(
\overrightarrow A \times \overrightarrow B = 0
\)
5) If two vectors \(\vec A\) and \(\vec B\) are parallel \(
\frac{{A_x }}
{{B_X }} = \frac{{A_y }}
{{B_y }} = \frac{{A_z }}
{{B_z }} = \text{constant}
\) or \(
\overrightarrow A \times \overrightarrow B = 0
\)
6) If two vectors are perpendicular to each other \(\theta=90^0\) then \(
\left| {\overrightarrow A \times \overrightarrow B } \right| = AB
\) (maximum)