Between two vectors, two distinct kinds of products are defined. One being a pure number is called the dot product while the other being a vector quantity is called vector product. In this section we will be learning more about dot product.

A product of two vectors $a$ and $b$ can be formed in such a way that the resultant is a scalar. The resultant is written as $a.b$ and called the dot product of $a$ and $b$.

1. Commutative law: The dot product of two vectors obeys commutative law, that is if $\vec{a}$ and $\vec{b}$ are any two vectors then

$\vec{a}.\vec{b}$ = $\vec{b}.\vec{a}$

2. Distributive law:The scalar product of two vectors obeys distributive law with respect to the addition of vectors, i.e., if $\vec{a}$, $\vec{b}$, $\vec{c}$ are any three vectors then

The dot product of matrices is defined for the matrices of same shape as the sum of the dot product of the vectors formed from the columns of one matrix with vectors formed from the corresponding column of the other matrix. The dot product of a real matrices is a real number, as is the dot product of real vectors. The dot product of the matrices A and B with the same shape is denoted by A.B, or <A,B>, just as the dot product of vectors.

For example, Let $A = [a^{1}, a^{2}, … a^{n}]$ and $B = [b^{1}, b^{2}, … b^{n}]$ be two matrices of the order $1 \times n$ and $n \times 1$ respectively.