State and explain the characteristics of vector product of two vectors. 

$(1)$ $\vec{a} \times \vec{b}=\vec{b} \times \vec{a}$

The vector product of two vector is not commutative but $\vec{a} \times \vec{b}=-\vec{b} \times \vec{a}$ is opposite to each other

However $|\vec{a} \times \vec{b}|=|\vec{b} \times \vec{a}|$

$(2)$ Scalar product act behave like reflection (taking image in mirror) $x \rightarrow-x, y \rightarrow-y$ and $z \rightarrow$ $-z$.

In reflection occurrence all components changes sign mean positive vector becomes negative.

So, $\vec{a} \times \vec{b} \rightarrow(-\vec{a}) \times(-\vec{b})=\vec{a} \times \vec{b}$

Hence, in reflection sign is not change in resultant.

$(3)$ Vector product obeys distributive law :

$\vec{a} \times(\vec{b}+\vec{c})=\vec{a} \times \vec{b}+\vec{a} \times \vec{c}$

$(4)$ For two non-zero vectors $\vec{a} \times \vec{a}=\overrightarrow{0}$

where $\overrightarrow{0}$ is vector of zero modulus

Here $\vec{a} \times \vec{a} =(a)(a) \sin 0^{\circ} \hat{n}$ $=\overrightarrow{0}$

( $\because$ Angle between $\vec{a}$ and $\vec{a}$ is $0^{\circ}$ )

Hence, condition of parallel or anti parallel of two non-zero vectors is that its vector product should be zero.

$(5)$ If two non-zero vector is perpendicular, then

$\vec{a} \times \vec{b} =a b \sin 90^{\circ} \hat{n}$

$=a b \hat{n}$

where $\hat{n}$ is unit vector in direction of $\vec{a} \times \vec{b}$.

$(6)$ Vector product for unit vector of cartesian co-ordinate system.