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Vector Addition

Learn all about vector addition and different methods like the parallelogram method and adding vectors graphically in the article below.

If you are a student of physics, then vector addition is extremely important for you to master. Most quantities used in physics, such as force or acceleration, are vectors. A vector is any object that has both magnitude and direction. So, the addition of vectors is different from scalar quantities that only have magnitude. 

In the following paragraphs, you will learn about different methods used to add vectors and determine the resultant. 

What is Vector Addition? 

A vector represents direction and magnitude. So, it is usually represented with an arrow above the quantity. For example, Vector a is written as a and so on. When you add two vectors, the resultant is written as a + b. 

There are some basic principles to keep in mind for addition of vectors: 

  • Vector addition is only possible when the vectors are of the same nature. For instance, you can add acceleration only with acceleration and not any other quantity. 
  • Scalars and vectors cannot be added. 

Basic Properties for Addition of Vectors 

Whether you are adding vectors graphically or using any other method, you must remember that algebraic addition is quite different from vector addition. There are some basic properties that will come handy to help you arrive at the correct answer. 

Vector Property 

Meaning 

Existence of Identity 

For any given vector, say a , 

a +0 = a

 

In this case,  0 is known as the additive identity. 

Associativity 

For three vectors, say a, b and c, 

(a+b)+c = a + (b+c)

Therefore, the order that you add the vectors in does not matter. 

Commutativity 

Addition of vectors is commutative. This means, 

 

a+b= b+a

Existence of inverse 

Every vector has an additive inverse. Therefore, 

 

a+ (-a)= 0

Adding Vectors Graphically 

Vector addition is simpler when you use graphical methods. There are three ways of adding vectors graphically: 

Addition of Vectors Using Components 

Vectors may be represented using cartesian coordinates. This system allows you to represent any point on a coordinate plane in the n-dimension. In simple terms, you represent them on the X and Y-axis. 

Any vector can be broken down into two components. These are the horizontal and vertical components. These components are at angle, let us say Φ. 

The components of a vector A are represented as, 

  • Ax , along the X-axis 
  • Ay , along the X-axis 

Mathematically, the magnitude of these components can be represented as 

  • Ax = A cosΦ
  • Ay = A sinΦ

Then, the magnitude of A can be calculated as

You can follow the same formula for vector addition. 

  • Let us say you have A and B
  • A = (a1,a2) which are the two components 
  • B = (b1, b2) 

So the resultant, let us call it R is calculated as follows: 

  • Rx = a1+b1
  • Ry = a2 +b2 
  • Then the final resultant is
Triangle Law of Addition of Vectors 

This is one of the most popular methods used for adding vectors graphically. It is also known as the head-to-tail method. 

According to this law of vector addition, you can join two vectors in a way where the head of the first one touches the tail of the second one. Then the hypotenuse or third side, which joins the tail of the first one and the head of the second one gives you the resultant. 

Let us consider two vectors, M and N: 

  • Join the head of vector M with the tail of vector N. They will form two sides of a triangle. 
  • Then draw a third line to join the tail of vector M with the head of N. This is third side of the triangle, P
  • Then, the vector addition is P= M+ N
Parallelogram Method 

The parallelogram method is another way of adding vectors graphically. Let us consider two vectors, M and N. Then follow a few simple steps: 

  • Create a parallelogram with M and N as the adjacent sides of the parallelogram. 
  • Remember, the adjacent sides of the parallelogram are always equal. 
  • The diagonal of the parallelogram gives you the result of addition of vectors, M and N. 
  • So the diagonal D=  M + N

In essence, the parallelogram method is similar to the triangle law of vector addition.

Conclusion 

The addition of vectors becomes simpler when you understand how to graphically represent them. As a revision, here are two principles that you should remember when adding vectors: 

  • You can only add vectors that represent the same quantity. For instance, you can add two vectors that represent acceleration. But not when one vector represents acceleration and the other represents force. 
  • Vector addition is both associative and commutative in nature. 
  • You cannot add vectors and scalars. 

The triangle law of vector addition is considered the first rule of vector addition.