Composition of Vectors

The process of adding two or more vectors or finding the resultant of these vectors is called composition of vectors. Since vectors have both magnitude and direction, they cannot be added by applying simple algebraic laws. Let us explain this by adding two displacement vectors in the following different cases.

When two vectors act in the same direction

Suppose a particle is displaced through $4$ $\text{m}$ and then $3$ $\text{m}$ due east. If these two displacement vectors are represented by $\overrightarrow{P}$ and $\overrightarrow{Q}$, then the sum $\overrightarrow{P}+\overrightarrow{Q}$ of the two displacement vectors will be the displacement of $7$ $\text{m}$ due east.

When two vectors act in opposite directions

Now, suppose the particle is displaced through $4$ $\text{m}$ due east and then $3$ $\text{m}$ due west, then the sum $\overrightarrow{P}+(-\overrightarrow{Q})$ of the two displacement vectors will be a displacement of $1$ $\text{m}$ due east.

When two vectors are inclined to each other

Suppose a particle is first displaced through $4$ $\text{m}$ due east and then through $3$ $\text{m}$ due north. These two displacements have been represented by $\overrightarrow{AB}$ and $\overrightarrow{BC}$ as shown in the figure given below. The measurements tell that the magnitude of displacement $\overrightarrow{AC}$ is $5$ $\text{m}$.

We can also measure the angle $\theta$ which gives the direction of $\overrightarrow{AC}$ due north of east. Therefore, instead of moving a distance of $4$ $\text{m}$ from $A$ to $B$ and then a distance of $3$ $\text{m}$ from $B$ to $C$ (a total distance of $4+3=7$ $\text{m}$), if the particle moves a distance of $5$ $\text{m}$ from $A$ to $C$ at an angle $\theta$ with $AB$, it would reach the same point.

Thus, the displacement of $5$ $\text{m}$ along $AC$ produces the same effect as the displacements of $4$ $\text{m}$ along $AB$ and $3$ $\text{m}$ along $BC$ together do. Hence, we can say that the displacement vector $\overrightarrow{AC}$ is the sum (or resultant) of the displacement vectors $\overrightarrow{AB}$ and $\overrightarrow{BC}$.

$\therefore\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$ If these vectors $\overrightarrow{AB}$, $\overrightarrow{BC}$ and $\overrightarrow{AC}$ are represented by vectors $\overrightarrow{P}$, $\overrightarrow{Q}$ and $\overrightarrow{R}$ respectively, then, $\overrightarrow{P}+\overrightarrow{Q}=\overrightarrow{R}$

From the discussion in the three cases, it follows that the vectors are not added algebraically but are added geometrically.

We have seen that when two vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ are inclined to each other as shown in the figure given below, their sum is given by $\overrightarrow{R}=\overrightarrow{P}+\overrightarrow{Q}$
If we observe it carefully, then we will find out that the vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ have been represented by the two sides $AB$ and $BC$ of a triangle taken in same order and their sum $\overrightarrow{R}$ is given by the side $AC$ taken in opposite side.