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.

## 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.

### 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:

1. The velocity imparted to the boat by its engine
2. 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.

### 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.

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