# Geometric addition of vectors

In physics and mathematics many times we need to add or subtract Vectors.

Although there are many ways of combining vectors or adding and subtracting vectors , The simplest and most straight forward method of combining vectors is by graphical method or geometrical method.

In Geometric vector addition we mainly use following two laws of vector addition:

## 1> **Law of triangle of vector addition** :

The law of triangle of vector addition states:

If , the two sides of a triangle represents two given vectors in magnitude and direction in same order , then third side drawn in opposite sense represents their vector sum.

For example:

Let there be two vectors and and the angle between them is as shown in the picture below:

Then , To find their sum( ) first of all we reposition the two vectors such that the head of vector exactly coincides with the tail of vector and then draw a vector from the tail of the vector to head of the vector , The newly drawn vector represents the vector b sum of vectors “a” and “b” , as shown in the figure below:

## 2> **Law of parallelogram of vector addition**:

It states that if two adjacent sides of a parallelogram represents two given vectors in magnitude and direction , then the diagonal starting from the intersection of two vectors represent their sum.

The example of law of parallelogram of vector addition is given in following picture:

**Properties of vector addition**:

Vector addition have following properties:

1> Commutative law:

This law states that the order of addition does not matter in vector addition. or,

2> Associative law:

This law states , when more than two vectors are added we can group them in any order, as we add them. Or,

**Subtraction of vectors**:

To define the subtaction of vectors first we need to define the negative vector of a vector.

The negative vector of vector is denoted by vector and is a vector with the same magnitude as of vector But with exactly opposite direction.

Adding has the same effect as subtracting , so we use the following formula to subtract a vector from another:

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