In this article, we will focus on vector addition. We will learn about the triangle law and parallelogram law along with the commutative and associative properties of vector addition.

**Table of content**

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## Triangle Law of Vector Addition

A vector \( \vec{AB} \), in simple words, means the displacement from point A to point B. Now, imagine a scenario where a boy moves from point A to B and then from point B to C. What is the net displacement made by him from point A to C?

This displacement is given by the vector \( \vec{AC} \), where

\( \vec{AC} \) = \( \vec{AB} \) + \( \vec{BC} \)

This is the Triangle law of Vector Addition.

**Browse more Topics under Vector Algebra**

- Basic Concepts of Vectors
- Components of a Vector
- Types of Vectors
- Scalar (or Dot) Product of Two Vectors
- Vector (or Cross) Product of Two Vectors
- Section Formula
- Projection of a Vector on a Line

**You can download Vector Algebra Cheat Sheet by clicking on the download button below**

### Let’s see how we can apply this law for vector addition:

If you have two vectors \( \vec{a} \) and \( \vec{b} \), then to add them you must position them in a manner that the initial point of one coincides with the terminal point of the other. This is shown in Fig. 2 (i) and (ii) below:

As can be seen in Fig. 2 (ii) above, the vector \( \vec{b} \) is shifted without changing its magnitude and direction, so that its initial point coincides with the terminal point of vector \( \vec{a} \). This helps us form the triangle ABC and the third side, AC, gives us the sum of the two vectors \( \vec{a} \) and \( \vec{b} \). Hence, from Fig. 2 (ii), we can write

\( \vec{AB} \) + \( \vec{BC} \) = \( \vec{AC} \)

Now, we know that \( \vec{AC} \) = – \( \vec{CA} \). Hence, from the above equation, we have

\( \vec{AB} \) + \( \vec{BC} \) = – \( \vec{CA} \)

Or, \( \vec{AB} \) + \( \vec{BC} \) + \( \vec{CA} \) = \( \vec{0} \)

Or, \( \vec{AA} \) = \( \vec{0} \)

which is also true because it is a zero vector as the initial and terminal points coincide, as shown below:

Now, as shown in Fig. 4 below, let’s construct a vector \( \vec{BC’} \), so that its magnitude is the same as vector \( \vec{BC} \), but the direction is opposite. Or,

\( \vec{BC’} \) = – \( \vec{BC} \)

Therefore, on applying the Triangle law of vector addition, we get

\( \vec{AC’} \) = \( \vec{AB} \) + \( \vec{BC’} \) = \( \vec{AB} \) + ( – \( \vec{BC’} \))

= \( \vec{a} \) – \( \vec{b} \)

The vector \( \vec{AC’} \) represents the difference between vectors \( \vec{a} \) and \( \vec{b} \).

## Parallelogram Law of Vector Addition

Now, let’s consider a slightly complex scenario. Imagine a boat in a river going from one bank to the other in a direction perpendicular to the flow of the river. This boat has two velocity vectors acting on it:

- The velocity imparted to the boat by its engine
- The velocity of the flow of the river.

When these two velocities simultaneously influence the boat, it starts moving with a different velocity. Let’s look at how we can calculate the resultant velocity of the boat.

To find the answer, let’s take two vectors \( \vec{a} \) and \( \vec{b} \) shown below, as the two adjacent sides of a parallelogram in their magnitude and direction.

Their sum, \( \vec{a} \) + \( \vec{b} \) is represented in magnitude and direction by the diagonal of the parallelogram through their common point. This is the Parallelogram law of vector addition.

Note: Using the Triangle law, we can conclude the following from Fig. 5

\( \vec{OA} \) + \( \vec{AC} \) = \( \vec{OC} \) Or,

\( \vec{OA} \) + \( \vec{OB} \) = \( \vec{OC} \) … since \( \vec{AC} \) = \( \vec{OB} \)

Hence, we can conclude that the triangle and parallelogram laws of vector addition are equivalent to each other.

## Properties of Vector Addition

### Property 1 – Commutative Property

For any two vectors \( \vec{a} \) and \( \vec{b} \),

\( \vec{a} \) + \( \vec{b} \) = \( \vec{b} \) + \( \vec{a} \)

**Proof**: To prove this property, let’s consider a parallelogram ABCD as shown below

Let, \( \vec{AB} \) = \( \vec{a} \) and \( \vec{BC} \) = \( \vec{b} \). Now, considering the triangle ABC and using the triangle law of vector addition, we have

\( \vec{AC} \) = \( \vec{a} \) + \( \vec{b} \)

In a parallelogram, the opposite sides are always equal. Hence, we have

\( \vec{AD} \) = \( \vec{BC} \) = \( \vec{b} \) and

\( \vec{DC} \) = \( \vec{AB} \) = \( \vec{a} \)

Next, considering the triangle ADC and using the triangle law of vector addition, we have

\( \vec{AC} \) = \( \vec{AD} \) + \( \vec{DC} \) = \( \vec{b} \) + \( \vec{a} \)

Hence, \( \vec{a} \) + \( \vec{b} \) = \( \vec{b} \) + \( \vec{a} \)

### Property 2 – Associative Property

For any three vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \),

(\( \vec{a} \) + \( \vec{b} \)) + \( \vec{c} \) = \( \vec{a} \) + (\( \vec{b} \) + \( \vec{c} \))

**Proof: **To prove this property, let’s look at two figures as given below

Let, the vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) be represented by \( \vec{PQ} \), \( \vec{QR} \), and \( \vec{RS} \), respectively. Now,

\( \vec{a} \) + \( \vec{b} \) = \( \vec{PQ} \) + \( \vec{QR} \) = \( \vec{PR} \)

Also, \( \vec{b} \) + \( \vec{c} \) = \( \vec{QR} \) + \( \vec{RS} \) = \( \vec{QS} \)

Hence, (\( \vec{a} \) + \( \vec{b} \)) + \( \vec{c} \) = \( \vec{PR} \) + \( \vec{RS} \) = \( \vec{PS} \)

And, \( \vec{a} \) + (\( \vec{b} \) + \( \vec{c} \)) = \( \vec{PQ} \) + \( \vec{QS} \) = \( \vec{PS} \)

Therefore, we conclude that

(\( \vec{a} \) + \( \vec{b} \)) + \( \vec{c} \) = \( \vec{a} \) + (\( \vec{b} \) + \( \vec{c} \))

Note: The associative property of vector addition enables us to write the sum of three vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) without using any brackets:

\( \vec{a} \) + \( \vec{b} \) + \( \vec{c} \)

Also, for any vector \( \vec{a} \), we have

\( \vec{a} \) + \( \vec{0} \) = \( \vec{0} \) + \( \vec{a} \) = \( \vec{a} \)

The zero vector is also called the additive identity for vector addition.

## Solved Examples for You

**Question 1: What is the Commutative property of vector addition?**

**Answer :** The Commutative property of vector addition states that for any two vectors \( \vec{a} \) and \( \vec{b} \),

\( \vec{a} \) + \( \vec{b} \) = \( \vec{b} \) + \( \vec{a} \)

Question: What is the Associative property of Vector addition?

Solution: The Associative Property of Vector addition states that for any three vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \),

(\( \vec{a} \) + \( \vec{b} \)) + \( \vec{c} \) = \( \vec{a} \) + (\( \vec{b} \) + \( \vec{c} \))

**Question 2: What is the Parallelogram law of vector addition?**

**Answer **: According to the Parallelogram law of vector addition, if two vectors \( \vec{a} \) and \( \vec{b} \) represent two sides of a parallelogram in magnitude and direction, then their sum \( \vec{a} \) + \( \vec{b} \) = the diagonal of the parallelogram through their common point in magnitude and direction.

**Question 4: What is meant by sum of two vectors?**

**Answer:** The sum of two or more vectors is known as the resultant. One can find the resultant of two vectors by using either the triangle method or the parallelogram method.

**Question 5: What is meant by magnitude of a vector?**

**Answer:** The magnitude of a vector refers to the length of the vector. The denotation of magnitude of the vector a is as ∥a∥. Formulas for the magnitude of vectors are available in two and three dimensions.

**Question 6: Explain the parallelogram law of vector?**

**Answer:** The Statement of Parallelogram law of vector addition is that in case the two vectors happen to be the adjacent sides of a parallelogram, then the resultant of two vectors is represented by a vector. Furthermore, this vector happens to be a diagonal whose passing takes place through the point of contact of two vectors.

**Question 7: What will be the result of two perpendiculars?**

**Answer:** When two force vectors are such that they are perpendicular to each other, their resultant vector is drawn so that the formation of a right-angled triangle takes place. In other words, the resultant vector happens to be the hypotenuse of the triangle.