Addition of Vectors

A vector  simply means the displacement from a point A to the point B. Now consider a situation that a girl moves from A to B and then from B to C in folloing fig.

The net displacement made by the girl from point A to the point C, is given by the vector `vec (AC)`  and expressed as
`vec (AC) = vec (AB) + vec (BC)`
This is known as the triangle law of vector addition.
There are two vectors `vec a` and `vec b` in following  first fig. then to add them, they are positioned so that the initial point of one coincides with the terminal point of the other in following second fig.


In above second fig. we have shifted vector `vec b` without changing its magnitude and direction, so that it’s initial point coincides with the terminal point of `vec a`. Then the vector `vec a +vec b` ,  represented by the third side AC of the triangle ABC, gives us the sum (or resultant) of the vectors `vec a` and `vec b` i.e. in triangle ABC in above fig .
`vec (AB) + vec (BC) = vec (AC)`
Now again, since  `vec (AC) = - vec (CA)`, from the above equation, we have
`vec (AB) + vec (BC) + vec (CA) = vec ("AA") =  vec 0`
This means that when the sides of a triangle are taken in order, it leads to zero resultant as the initial and terminal points get coincided in above third fig.
Now, construct a vector `vec (BC')` so that its magnitude is same as the vector `vec (BC)`,  but the direction opposite to that of it in above third fig.
`vec (BC)' = - vec (BC)`
Then, on applying triangle law from the Fig.
`vec (AC') = vec (AB) + vec (BC') = vec (AB) +  vec ((-BC)) = vec a - vec b`
The vector ` vec (AC')` is said to represent the difference of `vec a - vec b`.
If we have two vectors `vec a` and `vec b` represented by the two adjacent sides of a parallelogram in magnitude and direction in following fig.

then their sum `vec a + vec b` is represented in magnitude and direction by the diagonal of the parallelogram through their common point. This is known as the parallelogram law of vector addition.