Addition of Vectors

We cannot add two vectors directly like numbers to get the result as they have magnitude as well as direction. The addition of the scalar quantity is very simple, but it requires a different process to follow in the case of vectors. 

To know the difference and for better learning, let’s assume that a car is moving 10 miles to the north and then 10 miles to the south. We can easily evaluate the total distance traveled by car by adding these two numbers like 20 miles. But in the case of vector addition, the result is zero. 

The reason is that the north and south directions are opposite to each other, which is why they cancel out, and so the vector sum will be zero. This article provides a clear inference of the addition of 2 vectors, or we can say that “vector sum.”

Sum of Two Vectors

Let’s consider the two vectors \[\underset{u}{\rightarrow} + \underset{v}{\rightarrow}\]. We are going to add the corresponding components. Let’s write about the components of the vectors:

\[\underset{u}{\rightarrow} = (u_{1}, u_{2}) and \underset{v}{\rightarrow} = (v_{1}, v_{2})\]

When we do a summation of the above vectors, the result will be:

\[\underset{u}{\rightarrow} + \underset{v}{\rightarrow} = (u_{1} + v_{1}, u_{2} + v_{2})\]

The summation of two vectors can be called the resultant.

Vector Addition Formula

There are two types of vector addition methods, they are:

1. Triangle Law of Vectors

2. Parallelogram Law of Vectors

How do you Add Two Vectors?

Do you still wonder how to add vectors?

Here are some tips to remember for vector addition:

• The addition of vectors is accomplished geometrically but not algebraically.

• Vector quantities should behave as independent of each other quantities before the addition.

• From the vector addition, we only conclude the resultant of a number of vectors propagated on a body. 

• From vector addition, we obtain the resultant vector, which is not dependent on order of the summation of vectors as \[\underset{A}{\rightarrow} + \underset{B}{\rightarrow} = \underset{B}{\rightarrow} + \underset{A}{\rightarrow}\]

Triangle Law of Vector Addition

The vector addition is dependent when triangles are considered. Now, we need to find out how it works. 

Let’s assume that \[\underset{a}{\rightarrow}~and~\underset{b}{\rightarrow}\] are the two vectors.

Here, you ought to draw a line AB, which is called a tail with A and B with the head. Let’s draw a line BC, which allocates with B as the end and C as the head.

Let’s finish the triangle by drawing the line AC with A as the end and C as the crown. The sum of two vectors \[\underset{a}{\rightarrow}~and~\underset{b}{\rightarrow}\] is represented by the line AC.

(Image will be Uploaded soon) 

Mathematically, 

AC line =  \[\underset{a}{\rightarrow} + \underset{b}{\rightarrow}\]

We can calculate the magnitude of the AC line (\[\underset{a}{\rightarrow} + \underset{b}{\rightarrow}\])

\[\sqrt{a^{2} + b^{2} + 2ab cos \theta}\]

Here,

The magnitude of the vector \[\underset{a}{\rightarrow} = a\]

The magnitude of the vector \[ \underset{b}{\rightarrow} = b\]

θ is the angle covered by vector \[\underset{a}{\rightarrow} and ~Vector~\underset{b}{\rightarrow}\].

Consider that the resultant of the vectors make an angle of \[\Phi\] with \[\underset{a}{\rightarrow}\], then the expression will be:

\[tan \phi  = \frac{b sin\theta}{a + b cos\theta} = tan\frac{\theta}{2} \]

We need to learn this with the help of an example. Consider that we have two vectors with equal magnitude A, and θ is the angle between these two vectors. 

One can work out this formula to find the magnitude as well as the direction of the resultant. Suppose, B is the magnitude of the resultant, then the expression for this is:

\[B = \sqrt{A^{2}  + A^{2} + 2AAcos \theta} = 2A cos \frac{\theta}{2} \]

Consider that the resultant of the vectors make an angle of ф with  a; then the expression will be:

\[tan \phi  = \frac{A sin\theta}{A + A cos\theta} = tan\frac{\theta}{2} \]

Then, \[\phi  = \frac{\theta}{2} \]

Parallelogram Law of Vector Addition

We can also understand the concept of vector addition by using the law of parallelogram. 

The law of parallelogram states that “when two vectors are acting concurrently at a place (indicated by both sides of a parallelogram being marked from a point), then the result is given by the diagonal of that parallelogram with magnitude and direction passing through that same point.”

To make the law easier for understanding, consider two vector \[\underset{P}{\rightarrow} and \underset{Q}{\rightarrow}\]

These vectors are denoted by two adjacent sides of a parallelogram. They are indicated away from the point as per the figure given below.

(Image will be Uploaded soon)

The magnitude of the resultant can be stated as per the parallelogram law of vector addition.

\[(AC)^{2} = (AE)^{2} + (EC)^{2}\]

or ,⁡\[R^{2} = (P+Qcos⁡θ)^{2} (Qsin⁡θ)^{2}\]

or, \[R = \sqrt{P^{2}  + Q^{2}  + 2PQcos\theta}\]

Also, we need to determine the direction of the resultant vector:

\[tan \phi  = \frac{CE}{AE} =  \frac{Q sin\theta}{P + Q cos\theta} \]

\[\phi = tan^{-1}\frac{Q sin\theta}{P + Q cos\theta} \]

Vector Subtraction

The subtraction of two vectors is very much identical to addition. We need to assume that vector a is going to be subtracted from vector b. 

\[\underset{a}{\rightarrow} - \underset{b}{\rightarrow}\], We can write the expression like this.

We can also say that it is the addition of \[\underset{a}{\rightarrow} and \underset{-b}{\rightarrow}\]. That is why we can apply the same formula to calculate the resultant vector.

Vector Subtraction Formula is:

\[R = \overrightarrow{a} - \overrightarrow{b} = \sqrt{a^{2} + b^{2}  - 2abcos\theta}\]