Explain the kinds of multiplication operations for vectors.

(N/A) There are two kinds of multiplication of vectors:
$(i)$ Scalar product (Dot product):
If the product of two vector quantities results in a scalar,then the product is called a scalar product. This product is also known as the dot product.
The scalar product of two vectors $\vec{A}$ and $\vec{B}$ is denoted by $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta = AB \cos \theta$,where $A$ and $B$ are the magnitudes of $\vec{A}$ and $\vec{B}$ respectively,and $\theta$ is the angle between them.
$(ii)$ Vector product (Cross product):
If the product of two vector quantities results in a vector,then the product is called a vector product.
$A$ vector product is represented by placing a cross sign $(\times)$ between two vectors; hence,it is also called the cross product of vectors.
If $\theta$ is the angle between $\vec{A}$ and $\vec{B}$,then its vector product is $\vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin \theta \hat{n} = AB \sin \theta \hat{n}$,where $\hat{n}$ is the unit vector perpendicular to the plane formed by $\vec{A}$ and $\vec{B}$.