Here, you will learn triangle law of addition of vectors and parallelogram law of addition of vectors and properties of vector addition.

Let’s begin –

## Addition of Vectors

The vectors have magnitude as well as direction, therefore their addition is different than addition of real numbers.

Let \(\vec{a}\) and \(\vec{b}\) be two vectors in a plane, which are represented by AB and CD. Their addition can be performed in the following two ways :

## Triangle Law of Addition of Vectors

If two vectors can be represented in magnitude and direction by the two sides of a triangle, taken in order, then their sum will be represented by the third side in reverse order.

Let O be the fixed point in the plane of vectors. Draw a line segment \(\overrightarrow{OE}\) from O, equal and parallel to \(\overrightarrow{AB}\), which represents the vector \(\vec{a}\). Now from E, draw a line segment \(\overrightarrow{EF}\) equal and parallel to \(\overrightarrow{CD}\), which represents the vector \(\vec{b}\). Line segment \(\overrightarrow{OF}\) obtained by joining O and F represents the sum of vectors \(\vec{a}\) and \(\vec{b}\).

i.e. \(\overrightarrow{OE}\) + \(\overrightarrow{EF}\) = \(\overrightarrow{OF}\)

or \(\vec{a}\) + \(\vec{b}\) = \(\overrightarrow{OF}\)

This method of addition of two vectors is called **Triangle law of addition of vectors.**

## Parallelogram Law of Addition of Vectors

If two vectors be represented in magnitude and direction by the two adjacent sides of a parallelogram then their sum will be represented by the diagonal through the co-initial point.

Let \(\vec{a}\) and \(\vec{b}\) be vectors drawn from point O denoted by line segments \(\overrightarrow{OP}\) and \(\overrightarrow{OQ}\). Now complete the parallelogram OPRQ. Then the vector represented by the diagonal OR will represent the sum of the vectors \(\vec{a}\) and \(\vec{b}\).

i.e. \(\overrightarrow{OP}\) + \(\overrightarrow{OQ}\) = \(\overrightarrow{OR}\)

or \(\vec{a}\) + \(\vec{b}\) = \(\overrightarrow{OR}\)

This method of addition of two vectors is called **Parallelogram law of addition of vectors.**

## Properties of Vector Addition

(i) \(\vec{a}\) + \(\vec{b}\) = \(\vec{b}\) + \(\vec{a}\) (Commutative)

(ii) (\(\vec{a}\) + \(\vec{b}\)) + \(\vec{c}\) = \(\vec{a}\) + (\(\vec{b}\) + \(\vec{c}\)) (associativity)

(iii) \(\vec{a}\) + 0 = 0 + \(\vec{a}\) (additive identity)

(iv) \(\vec{a}\) + (-\(\vec{a}\)) = 0 (additive inverse)