Explain the kinds of multiplication operations for vectors.

There are two kinds of multiplication of vectors :

$(i)$ Scalar product (Dot product) :

If the product of two vector quantities results into a scalar then the product called a scalar product. This product is also known as dot product.

The scalar products of two vectors $\vec{A}$ and $\overrightarrow{\vec{B}}$ is denoted by, $\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}=|\overrightarrow{\mathrm{A}}||\overrightarrow{\mathrm{B}}| \cos \theta$

$=A B \cos \theta$ where $A$ and $B$ are the magnitudes of $|\vec{A}|$ and $|\vec{B}|$ respectively and $\theta$ is the angle between $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$

$(ii)$ Vector product (Cross product) :

If the product of two vector quantities results into a vector then the product is called a vector product.

A vector product is represented by keeping cross $\operatorname{sign}(\times)$ between two vectors hence it is also called cross product of vectors.

Suppose $\theta$ is the angle between $\vec{A}$ and $\vec{B}$, then its vector product is, $\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}=|\overrightarrow{\mathrm{A}} \| \overrightarrow{\mathrm{B}}| \sin \theta \cdot \hat{n}$

$=\mathrm{AB} \sin \theta \hat{n}$ where $\hat{n}$ is the unit vector in the direction perpendicular to the plane formed by $\overrightarrow{\mathrm{A}}$ and $\vec{B}$ and $A$ and $B$ are the magnitude of $\vec{A}$ and $\vec{B}$ respectively.