A vector is a quantity known for having both magnitude and direction.It is an element present in the vector space. They are referred based on the number of coordinates like 2- dimensional vector, 3-dim,.... n-dimensional. There are many vector formulas based on its properties. Some of them are, 

Magnitude:

The vector is represented by, A = a $\vec{i}$ + b $\vec{j}$, the magnitude is given by |A| = $\sqrt{a^2 + b^2}$.

Vector addition:
If two vectors $\vec{A}$ and $\vec{B}$ that represent the force acts in the same direction, then the resultant R gives out the sum of vectors.
$\vec{R} = \vec{A} + \vec{B}$
Vector Addition

Vector subtraction: 
If two vectors $\vec{A}$ and $\vec{B}$ that represent the forces acts in the opposite direction, then the resultant R gives out the vector subtraction.

$\vec{R} = \vec{A} - \vec{B}$
Vector Subtraction


Angle between two vectors :
If $\theta$ is the angle between two vectors $\vec{A}$ and $\vec{B}$ then angle between two vectors is given by,

$\Theta = cos^{-1}\frac{a.b}{|a||b|}$

Dot Product

Where, $\theta$ is the angle between the vectors a and b.

Below are given some examples on vector:

Question 1: If $\vec{a}$ = 3 $\vec{i}$ + 2 $\vec{j}$ - 7 $\vec{k}$ and $\vec{b}$ = 2 $\vec{i}$ - 5 $\vec{j}$ + 3 $\vec{k}$ acts in the same direction, Calculate its resultant force.
Solution:
Given: $\vec{a}$ = 3 $\vec{i}$ + 2 $\vec{j}$ - 7 $\vec{k}$  
$\vec{b}$ = 2 $\vec{i}$ - 5 $\vec{j}$ + 3 $\vec{k}$

$\vec{a}$ + $\vec{b}$ = 3 $\vec{i}$ + 2 $\vec{j}$ - 7 $\vec{k}$ + ( 2 $\vec{i}$ - 5 $\vec{j}$ + 3 $\vec{k}$)
$\vec{a}$ + $\vec{b}$ = 5 $\vec{i}$ - 3 $\vec{j}$ - 4 $\vec{k}$.

The Resultant force, R = 5 $\vec{i}$ - 3 $\vec{j}$ - 4 $\vec{k}$.

Question 2: If vector 7 $\vec{i}$ - 2 $\vec{k}$ acts opposite to 6 $\vec{i}$ + 2 $\vec{j}$ + 3 $\vec{k}$. Calculate the Resultant force.
Solution:
Let $\vec{a}$ = 7 $\vec{i}$ - 2 $\vec{k}$  
$\vec{b}$ = 6 $\vec{i}$ + 2 $\vec{j}$ + 3 $\vec{k}$

$\vec{a}$ - $\vec{b}$ = 7 $\vec{i}$ - 2 $\vec{k}$ - (6 $\vec{i}$ + 2 $\vec{j}$ + 3 $\vec{k}$)
$\vec{a}$ - $\vec{b}$ = $\vec{i}$ - 2 $\vec{j}$ - 5 $\vec{k}$

The Resultant force, R =  $\vec{i}$ - 2 $\vec{j}$ - 5 $\vec{k}$.