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 $\overrightarrow{A}$ and $\overrightarrow{B}$ and the angle between them is $\theta$ as shown in the picture below:

Then , To find their sum( $\overrightarrow{A} + \overrightarrow{B}$ )  first of all we reposition the two vectors such that the head of vector $\overrightarrow{A}$ exactly coincides with the tail of vector $\overrightarrow{B}$ and then draw a vector from the tail of the vector $\overrightarrow{A}$ to head of the vector $\overrightarrow{B}$ , 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:

1> Commutative law:

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

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

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,

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

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 $\overrightarrow{A}$ is denoted by vector $- \overrightarrow{A}$ and is a vector with the same magnitude as of vector $\overrightarrow{A}$ But with exactly opposite direction.

Adding $- \overrightarrow{b}$ has the same effect as subtracting $\overrightarrow{b}$ , so we use the following formula to subtract a vector from another:

$\overrightarrow{a} - \overrightarrow{b} = \overrightarrow{a} + ( - \overrightarrow{b} )$

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