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## Vector Addition- Analytical Method

Adding two or more vectors is called vector addition. What is a vector? Any quantity which has magnitude along with direction is a vector. Some of the examples for vector quantities are displacement, velocity, acceleration, force, pressure etc. Mathematical operations can be performed between two or more vectors. In this article, we will learn about the vector addition of two quantities using the analytical methods.

**What is vector addition?**The process of adding two or more vectors is called vector addition. Depending on the direction of the vector, vector addition is categorised into two types. They are –

**Parallelogram law of vector addition**

**Triangular law of vector addition.**

Consider a vector `bar A and bar B`

**Parallelogram law of vector addition**If two vectors are arranged head to head or tail to tail then, the parallelogram law of vector addition is carried out.

**Statement“If two vectors are considered to be the adjacent sides of a parallelogram, then the resultant of two vectors is given by the vector that is a diagonal passing through the point of contact of two vectors.”**

**Method**Step-wise vector addition of two vectors using Parallelogram law of vector addition is given below-

**Step 1:**Consider two vectors; `bar A and bar B`

**Step 2:**Bring the tail of `bar A` to the tail of `bar B`. Here the direction of vectors is not changed.

**Step 3:**Draw a lines parallel to `bar A` and `bar B` with the same magnitude, in a way to complete parallelogram.

**Step 4:**Join the point o and r by a straight line with an arrow pointing towards the r. This is diagonal to the parallelogram.

And this is the resultant vector `bar R, bar A + bar B = bar R`

**Triangular law of vector addition.**

If two vectors are arranged head to tail the traingular law of vector addition is carried out.

**Statement:**

**“When two vectors are represented by two sides of a triangle in magnitude and direction were taken in the same order then the third side of that triangle represents in magnitude and direction the resultant of the vectors.”**

**Method:**Step-wise vector addition of two vectors using Triangular law of vector addition is given below-

**Step 1:**Consider two vectors, `bar A and bar B`

**Step 2:**Bring the head of `bar A` to the tail of `bar B`. Here the direction of vectors is not changed.

**Step 3:**Join the tail of `bar A`to the head of `bar B`by a straight line with an arrow pointing towards the head of `bar B`

This new vector is the resultant vector `bar C, bar A + bar B = bar C`

**Why vector addition is important?**In physics, vector quantities like force interact with each other and produce a resultant effect on the objects upon which they are applied. Since the impact of all these forces is taken into consideration when finding the nature of motion of the system, so, in order to find the resultant of these forces, operations such as addition, subtraction and multiplication are required to be performed on these forces.