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 $b$.

Dot Product
The dot product of two vectors $\vec{a}$ and $\vec{b}$, with magnitude $a$ and $b$ respectively is defined by the real number $ab \cos \theta$

i.e., $\vec{a}.\vec{b}$ = $ab \cos \theta$
where $\theta$ is the angle between the direction of $\vec{a}$ and $\vec{b}$.

The value of $\theta$ is restricted to the interval $0 \leq \theta \leq \pi$.
It makes no difference whether $\theta$ or $-\theta$ is chosen as

$\cos \theta$ = $\cos (-\theta)$

Since the product of two vectors are scalar, it is also known as scalar product. The usual notations for dot product of two vectors are given below

$\vec{a}.\vec{b}$ or $(\vec{a},\vec{b})$

$\vec{a}.\vec{b}$ read as $\vec{a}$ dot $\vec{b}$
The angle $\theta$ between two vectors $a$ and $b$ is given by

$\cos \theta$ = $\frac{a.b}{|a|.|b|}$ with $0 \leq \theta \leq \pi$.

The examples of dot products are:

1. Power is the dot product of force ($\vec{F}$) and velocity (\vec{v}).

$P = \vec{F}.\vec{v}$

2. Work is the dot product of force ($\vec{F}$) and displacement (\vec{r}).

$W = \vec{F}.\vec{r}$

3. Electric flux is the dot product of intensity of electric filed ($\vec{E}$) and normal area (\vec{A}).
$\phi = \vec{E}.\vec{A}$
If vectors $a$ and $b$ parallel, then the angle between them is zero.

i.e., $\theta = 0$ : $\cos 0 = 1$

$\vec{a}.\vec{b}$ = $ab \cos \theta$

= $|a|.|b|$ and in particular $a.a$ = $|a|^{2}$
The following are the properties of dot product.

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

$\vec{a}.(\vec{b}+\vec{c}$ = $\vec{a}.\vec{b} + \vec{a}.\vec{c}$

3. The scalar product of two vectors obeys associative law with respect to a scalar, i.e., if x is a scalar, then

$(x\vec{a}).\vec{b}$ = $x(\vec{a}.\vec{b})$ = $\vec{a}.(x \vec{b})$
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.

The dot products of the matrices is

$A.B$ = $a^{1}b^{1}+a^{2}b^{2}+......+a^{n}b^{n}$

= $\sum^{n}_{ i=1}a^{i}b^{i}$
The quantity what we obtain from the dot product is called the scalar projection of one vector onto another.
To obtain the scalar projection of the vector A onto the vector B, we can use the following formula

$proj_{B}A$ = $\frac{A.B}{|B|}$
The following are the examples of dot product

Solved Examples

Question 1: Find the dot product of $\vec{a}$ and $\vec{b}$ where, $\vec{a}$ =2i+3j+4k and $\vec{b}$ = 4i+5j+3k
Solution:
 
Given $\vec{a}$ = 2i+3j+4k and $\vec{b}$ = 4i+5j+3k

Dot product of both the vectors is

$\vec{a}$.$\vec{b}$ = (2i+3j+4k).(4i+5j+3k)

= $(2\times 4)i.i+(3\times 5)j.j+(4\times 3)k.k$

= $8+15+12$

= $35$
 

Question 2: Find the dot product of $\vec{a}$ and $\vec{b}$ where, $\vec{a}$ =i+j+k and $\vec{b}$ = 2i+4j+1k
Solution:
 
Given $\vec{a}$ = i+j+k and $\vec{b}$ = 2i+4j+1k

Dot product of both the vectors is

$\vec{a}$.$\vec{b}$ = (i+j+k).(2i+4j+1k)

= $(1\times 2)i.i+(1\times 4)j.j+(1\times 1)k.k$

= $2+4+1$

= $7$
 

Question 3: Find the dot product of $<4,5>$ and $<2,3>$
Solution:
 
$<4,5>$ . $<2,3>$ = $4(2) + 5(3)$

= $8 + 15$ 

= $23$