# VECTOR ADDITION AND SUBTRACTION IN PHYSICS

Vector addition and subtraction in physics are done by graphical methods or components.

The idea behind the vector addition can be described with an example. Suppose that a person is standing on a table at point O. He walks along a straight line from point O to point P. It just so happens that another person pushes the table during that same amount of time as first person walks from point O to P.  So the point P will have moved exactly the same way as the table. If these two motions take place simultaneously, then the observer outside the table will see that the person moves from point O to S along a straight line.

$${\vec { OS } }={\vec { OP }}+{ \vec { PS } }$$

There are various ways of adding vectors. Suppose there are two vectors $${ \vec { A } }$$ and $${ \vec {B}}$$. Tail of vector $${ \vec {B}}$$ can be placed to the head of $${ \vec { A}}$$. The net result is the arrow drawn from the tail of the first vector ($${ \vec { A } }$$) to the head of the second vector ($${ \vec {B}}$$), $${ \vec { A}}+{ \vec { B}}={ \vec { C }}$$ . This method is known as triangle (or head to tail) method for vector addition. According to this method, it doesn’t matter in which order the vectors are added. So placing tail of $${ \vec {B}}$$ to head of $${ \vec {A}}$$ or tail of $${ \vec {A}}$$ to head of $${ \vec {B}}$$ gives same resultant vector.

## Parallelogram Method of Vector Addition

A second way to add two vectors is the parallelogram method. In this method, the two vectors tails are brought to same point and a parallelogram is drawn using these two vectors as adjacent sides. The resultant is the diagonal vector $${ \vec {C}}$$ and it is the sum vector of $${ \vec {A}}$$ and $${ \vec {B}}$$. This method is equivalent to head to tail method.

It is not important at which order the vectors are added. That is,

$${ \vec {A}}+{ \vec {B}}={ \vec {B}}+{ \vec{A}}$$ (commutative law)

First, each vector that going to be added are resolved into its components. Then x-components of the resolved vectors are summed. The same is true for the y-components and z-components.

So the addition of two vectors in 2-dimensional space is shown below, $$\vec { R } =\vec { A } +\vec { B }$$ implies that

$${ R }_{ x }={ A }_{ x }+{ B }_{ x }$$
$${ R }_{ y }={ A }_{ y }+{ B }_{ y }$$
The x-component of resultant R vector is equal to sum of the x-components of vectors which are being added. This situation is similar for y-components also. x-components and y-components are not added together.

Here in the figure, x and y components of vectors are all positive.  Above equation are also valid for components with any signs.

The above procedure can be extended to find the sum of any number of vectors.

$$\vec {R}=\vec {A}+\vec {B}+\vec {C}+\vec {D}\dots$$

## Subtraction of Vectors

What is the negative of a vector $${ \vec {A}}$$ ? The negative of a vector $${ \vec {A}}$$ is a vector with same magnitude as $${ \vec {A}}$$ but flip over 180 °.

If $${ \vec {-A}}$$ vector is added to $${ \vec {A}}$$, the result is 0.  The subtraction of vectors is equal to the sum of the first vector plus the negative of the second.