Define vector product. Explain the properties of a vector product with two examples.

Vector product: The vector product of any two vectors, is a vector, whose magnitude is equal to the product of the magnitudes of the two vectors and sine of angle between them & direction of vector product is perpendicular to the plane containing the two vectors.
$\begin{array}{l}(\therefore \mathrm{AQ}=\mathrm{AC}+\mathrm{CQ})\\ \therefore \overline{\mathrm{P}}\times \overline{\mathrm{Q}}=\mathrm{PQsin}\mathrm{\theta }\widehat{\mathrm{n}}\Rightarrow |\overline{\mathrm{P}}\times \overline{\mathrm{Q}}|=\mathrm{PQsin}\mathrm{\theta }\end{array}$
Where $\hat{\mathrm{n}}$ is a unit vector.
Properties:
1. Cross product does not obey commutative law:
$\overline{\mathrm{P}}\times \overline{\mathrm{Q}}\ne \overline{\mathrm{Q}}\times \overline{\mathrm{P}}\text{ but }(\overline{\mathrm{P}}\times \overline{\mathrm{Q}})=-(\overline{\mathrm{Q}}\times \overline{\mathrm{P}})$
2. Cross product obey’s distributive law:
$\overline{\mathrm{P}}\times (\overline{\mathrm{Q}}+\overline{\mathrm{R}})=(\overline{\mathrm{P}}\times \overline{\mathrm{Q}})+(\overline{\mathrm{P}}\times \overline{\mathrm{R}})$
3. The magnitude of cross product of two parallel vectors is zero ($\mathrm{\theta }=0°$)
$\Rightarrow |\overline{\mathrm{P}}\times \overline{\mathrm{Q}}|=\mathrm{PQsin}{0}^{\circ }=0$
4. The magnitude of cross product of two perpendicular vector is ($\mathrm{\theta }$ = 90°)
$\Rightarrow |\overline{\mathrm{P}}\times \overline{\mathrm{Q}}|=\mathrm{PQsin}{90}^{\circ }=\mathrm{PQ}$
$\text{ 5. If }\hat{\mathrm{i}},\hat{\mathrm{j}},\hat{\mathrm{k}}\text{ are unit vectors along }\mathrm{x},\mathrm{y},\mathrm{z}\text{ axis }$
$\begin{array}{l}\hat{\mathrm{i}}\times \hat{\mathrm{i}}=\hat{\mathrm{j}}\times \hat{\mathrm{j}}=\hat{\mathrm{k}}\times \hat{\mathrm{k}}=0\\ \widehat{\mathrm{i}}\times \widehat{\mathrm{j}}=\widehat{\mathrm{k}}\\ \widehat{\mathrm{j}}\times \widehat{\mathrm{k}}=\widehat{\mathrm{i}}\\ \widehat{\mathrm{k}}\times \widehat{\mathrm{i}}=\widehat{\mathrm{j}}\end{array}$
Examples of vector product of two vectors :
a) Torque (or) moment of force $\left(\overrightarrow{\mathrm{\tau }}\right):$
$\therefore \overrightarrow{\mathrm{\tau }}=\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{F}}$
b) Angular momentum (L) $\overrightarrow{\mathrm{L}}:$
$\therefore \overrightarrow{\mathrm{L}}=\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{P}}$