Addition of Vectors

Top
Download PDF

Vector Addition

We cannot add two vectors to get the result as they have magnitude as well as direction. The addition of the scalar quantity is very simple, but it is a little complicated in the case of vectors. 

To know the difference and better learning, let’s assume that a car is moving 10 miles to the north and 10 miles to the south. We can easily tell 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 \[\overrightarrow{u}\] and \[\overrightarrow{v}\]. We are going to add the corresponding components. Let’s write about the components of the vectors:

\[\overrightarrow{u}\] = ⟨u1,u2⟩ and \[\overrightarrow{v}\] = ⟨v1,v2

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

\[\overrightarrow{u}\] + \[\overrightarrow{v}\] =⟨u1+v1 , u2+v2

The summation of two vectors can be called as 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 the summation of vectors as \[\overrightarrow{A}\] + \[\overrightarrow{B}\] =  \[\overrightarrow{B}\] + \[\overrightarrow{A}\] .


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 \[\overrightarrow{a}\]  and \[\overrightarrow{b}\] are the two vectors.

Here, you ought to draw a line AB, which is called \[\overrightarrow{a}\] a tail with A and \[\overrightarrow{b}\]  with the head. Let’s draw a line BC, which allocates \[\overrightarrow{b}\] 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 \[\overrightarrow{a}\] & \[\overrightarrow{b}\] is represented by the line AC.

[Image will be uploaded soon]

Mathematically, 

AC line = \[\overrightarrow{a}\] + \[\overrightarrow{b}\]

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

\[\sqrt{a^{2}+b^{2}+2abcos⁡θ}\]

Here,

The magnitude of the vector = a

The magnitude of the vector = b

θ is the angle covered by vector  \[\overrightarrow{a}\] & vector \[\overrightarrow{b}\]

Consider that the resultant of the vectors make an angle of ф with \[\overrightarrow{a}\] ; then the expression will be:

tan⁡ф=\[\frac{bsin⁡θ}{a+bcos⁡θ}\]

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 on 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= \[\sqrt{A^{2}+A^{2}+2AAcos⁡θ}\]= 2Acos⁡\[\frac{θ}{2}\]

Consider that the resultant of the vectors make an angle of ф with \[\overrightarrow{a}\]; then the expression will be:

tan⁡ф=\[\frac{A sin⁡θ}{A+A cos⁡θ}\] = tan tan\[\frac{θ}{2}\]

Then, ф = \[\frac{θ}{2}\]


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 \[\overrightarrow{P}\] and \[\overrightarrow{Q}\]. 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 ,⁡R2 = (P+Qcos⁡θ)2(Qsin⁡θ)2

or, R = \[\sqrt{(P^{2}+Q^{2}) + 2PQcos⁡θ}\]

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

tantanO  = CE⁡/AE = Qsin⁡θ/(P + Qcos⁡θ)

O = tan-1⁡[Qsin⁡θ/(P + Qcos⁡θ)]


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.

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

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

Vector Subtraction Formula is:

R = \[\overrightarrow{a}\]-\[\overrightarrow{b}\] = \[\sqrt{(a^{2}+b^{2}) - 2abcos⁡θ}\]

Here, \[[-\overrightarrow{b}]\] = \[\overrightarrow{b}\] set aside in the direction

FAQs (Frequently Asked Questions)

Q1. How do you define the term used for the summation of two vectors?

Ans: The summation of two vectors is known as the resultant. Here, we notice that the two vectors are the same in magnitude and direction to each other. When the magnitude and direction of two vectors are different, then they are parallel to each other.

Q2. What is the possibility of adding three vectors?

Ans: Yes, we can add three vectors. As a result, we will get a unit vector. The answer will be ‘no’ when two unit vectors are along the coordinate axes. This is because the magnitude of the unit vectors is identical but possess dissimilar directions.

Q3. Is it possible that a vector can have zero components?

Ans: Yes. 

Along a line, a vector can have zero components. It is also possible that the vector possesses a non-zero magnitude. A vector can have a non-zero x-component along with the zero y-component.

Q4. How can a student get help from Vedantu in Physics?

Ans: Students are most welcome to join Vedantu classes. We help our students with genuine and qualitative Physics content to make the learning easier. Students of Vedantu can have the freedom to enjoy their lessons and practice the questions as much as possible to secure good marks in the board exam.

Share this with your friends
SHARE
TWEET
SHARE
SUBSCRIBE