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

Geometric addition of vectors

Geometric addition of vectors

 

 

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:

 

Geometric addition of vectors

Geometric addition of vectors

 

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:

Geometric addition of vectors

Geometric addition of vectors

 

 

 

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,

\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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