The physical quantities which have both magnitude and direction are called vector quantities.They are added and subtracted according to simple laws. In this section we will be learning more about different methods of vector addition.

Two vectors are added if they are of the same length that is have the same number of elements; the sum of two vectors is the vector whose elements are the sums of the corresponding elements of addends. Vector with same number of elements are said to be conformable for addition.
Let $\vec{A}$ and $\vec{B}$ be the two vectors acting on the same direction. To find their resultant, coincide the tail of second vector ($\vec{B}$) on the head of the first vector ($\vec{A}$). That is

$\vec{R}$ = $\vec{A} + \vec{B}$

Laws of vector addition

1. Vector sum is Commutative:

$\vec{A}+\vec{B}$ = $\vec{B}+\vec{A}$

2. Vector sum is Associative:

$\vec{A}+(\vec{B}+\vec{C})$ = $\vec{B}+(\vec{C}+\vec{A})$ = $\vec{C}+(\vec{B}+\vec{A})$

3. Vector sum is Distributive:

$m(\vec{A}+\vec{B})$ = $m\vec{A}+m\vec{B}$
If two vectors can be represented magnitude and direction by the two sides of a triangle taken in the same order, then the resultant is represented in direction and magnitude by the third side of the triangle taken in the opposite sense.

If we have to find $\vec{A}+\vec{B}$, then draw $\vec{A}$ and then $\vec{B}$ such that the tail of $\vec{B}$ coincides with vector $\vec{A}$. Then join the head and tail of $\vec{B}$ and $\vec{A}$ by joining head to head and tail to tail . This vector will represent $\vec{A}+\vec{B}$.   

Polygon

If two vectors acting simultaneously at a point can be represented in direction and magnitude by two adjacent sides of a parallelogram then their resultant is given completely in direction and magnitude by that diagonal of a parallelogram which passes through their point of intersection.

In this method the tail of the two vectors are kept at same place. The resultant vector is obtained by that diagonal of the parallelogram which passes through the point if intersection of two vectors. Here the parallelogram constructed with sides $\vec{A}$ and $\vec{B}$.

Parallelogram

It states that if a number of vectors be represented in direction and magnitude by the sides of a polygon taken in order then their resultant is given in magnitude and direction by the closing side of the polygon taken in the opposite order. 

A polygon is formed by drawing $\vec{A}$, $\vec{B}$, $\vec{C}$ and $\vec{D}$ such that the tail of $\vec{B}$ coincide with head of $\vec{A}$, the head of $\vec{B}$ is coincide with tail of $\vec{C}$ and so on. The closing side of polygon gives the sum of all the vectors.

Polygon Picture
A vector quantity can be represented by an arrow-tipped line segment. The length of the line, drawn to scale represents the magnitude of the quantity. The direction of the arrow indicates the direction of the quantity. This arrow-tipped line segment represent a vector.

By using graphical method the vectors are added by placing the tail of one vector at the head of the other vector. The resultant R, represents the sum of $\vec{A}$ and $\vec{B}$.

The order of addition dose not matter in this case and the vector can also have different directions.   

Vector by Graphical Method

when vectors to be added are not perpendicular, the method of addition by components described below can be used.

To add two or more vectors A, B, C ….. by the component method, the following procedure can be used

1. Resolve the initial vectors into components in the $x$, $y$, and $z$ directions.

2. Add the components in the $x$ direction to give $R_{x}$, add the components in the $y$ direction to give $R_{y}$, and add the components in the $z$ direction to give $R_{z}$. That is the magnitude of $R_{x}$, $R_{y}$ and $R_{z}$ are given by,

$R_{x}$ = $A_{x}+B_{x}+C_{x}+....$

$R_{y}$ = $A_{y}+B_{y}+C_{y}+....$

$R_{z}$ = $A_{z}+B_{z}+C_{z}+....$
3. Calculate the magnitude of the resultant R from its components $R_{x}$, $R_{y}$ and $R_{z}$ using the polygon theorem:

$R$ = $\sqrt{R^{2}_{x}+R^{2}_{y}+R^{2}_{z}}$
The following are the examples for vector addition.

Solved Examples

Question 1: Add $3\vec{i}-2\vec{j}+4\vec{k}$ with $5\vec{i}+3\vec{j}-2\vec{k}$
Solution:
 
The given vectors are 

$\vec{A}$ = $3\vec{i}-2\vec{j}+4\vec{k}$

$\vec{B}$ = $5\vec{i}+3\vec{j}-2\vec{k}$

The resultant $\vec{R}$ = $\vec{A}$ + $\vec{B}$

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

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

= $8\vec{i}+\vec{j}+2\vec{k}$
 

Question 2: Find the sum of the vectors $2\vec{i}+3\vec{j}+6\vec{k}$ and $5\vec{i}+2\vec{j}+3\vec{k}$ and hence find the magnitude.
Solution:
 
The given vectors are 

$\vec{A}$ = $2\vec{i}+3\vec{j}+6\vec{k}$

$\vec{B}$ = $5\vec{i}+2\vec{j}+3\vec{k}$

The resultant $\vec{R}$ = $\vec{A}$ + $\vec{B}$

= $2\vec{i}+3\vec{j}+6\vec{k}$ + $5\vec{i}+2\vec{j}+3\vec{k}$

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

= $7\vec{i}+5\vec{j}+9\vec{k}$  

Magnitude $|R|$ = $\sqrt{7^{2}+5^{2}+9^{2}}$

= $\sqrt{49 + 25 + 81}$

= $\sqrt{155}$

= $12.45$