# Triangle Law of Addition of Vectors | Parallelogram Law

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 –

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.

(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)