Laws of Vector Addition

Vectors refer to objects that can have both direction and magnitude. If there are any two vectors having the same magnitude and direction, then these two vectors are regarded as the same. These are geometrical entities that are represented by a line and an arrow. This arrow points towards the direction of the vector whereas the length of the line represents the magnitude of the vector. Therefore, these arrows have an initial point and a terminal point. Vectors represent physical quantities like velocity, displacement and acceleration.

Let’s go through the law of vector addition pdf. 

A vector has magnitude (that is the size) and direction. The length of the line or the arrow given above shows its magnitude and the arrowhead points in the direction. Now, we can add two vectors by simply joining them head-to-tail, refer to the diagram given below for better understanding. Also, it doesn't matter in which order the two vectors are added, we get the same result anyway.

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Notation

• A vector can often be written in bold, like a or b.

• A vector can also be written as the letters of its tail and head with an arrow above it, as shown on the right side. 

In this article, we will discuss the vector addition, triangle law of vector addition, parallelogram law of vector addition, and the law of vector addition pdf.

What is the Addition of Two Vectors?

In general terms, it says you can add two vectors and the result will be a vector. For example, let’s consider 

V=R²={(a,b)|a,b∈R} 

Then for vector(v₁)=(x₁ , y₁), (v₂)=(x₂ , y₂) we have, 

v₁+v₂=(x₁ + x₂ , y₁ + y₂)

What are the Properties of Vector Addition?

The addition of vectors satisfies two important properties.

1. The Commutative law states that the order of addition doesn't matter, that is: A+B is equal to B+A.

2. The Associative law states that the sum of three vectors does not depend on which pair of vectors is added first, that is (A+B)+C=A+(B+C).

Triangle Law of Vector Addition

Let’s discuss the triangle law of vector addition in the law of vector addition pdf. Suppose, we have two vectors namely A and B as shown.

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Now, the method to add these two vectors is very simple.  We need to simply place the head of one vector over the tail of the other vector as shown in the figure below.

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Now after this, we need to join the other endpoints of both the vectors together as shown below,

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The resultant of the given vectors (A and B) is given by a vector C which represents the sum of vectors A and B that is, C = A+B

From the law of vector addition pdf, vector addition is commutative in nature i.e.

If C=A+B; then we can write C = B+A

Similarly, if we need to subtract both the vectors using the triangle law then we simply reverse the direction of any vector and then add it to another one as shown below.

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Now we can mathematically represent this as C = A-B (as they are in opposite directions). This is the Triangle Law of Vector Addition.

Parallelogram Law of Vector Addition

The Mathematics law of vector addition named the parallelogram law of vector addition generally states that the sum of the squares of the length of the four sides of a parallelogram is equal to the sum of the squares of the length of the two diagonals of the parallelogram. In Euclidean Geometry, it is necessary that the parallelogram must have equal opposite sides.

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If ABCD is a parallelogram, then AB is equal to DC and AD is equal to BC. Then according to the definition of the parallelogram law, it is stated as,

2(AB)² + 2 (BC)² =  (AC)² + (BD)²

In case, the parallelogram is a rectangle, then the law can be stated as,

2(AB)² + 2 (BC)²=  (AC)²

This is because, in a rectangle, two diagonals are of equal lengths. i.e., (AC=BD)

Parallelogram Law of Addition of Vectors Procedure

The steps for the parallelogram law of the addition of vectors are given below:

Step 1) Draw a vector using a suitable scale in the direction of the vector.

Step 2) In this step you need to draw the second vector using the same scale from the tail of the first given vector.

Step 3) Now, you need to treat these vectors as the adjacent sides and then complete the parallelogram.

Step 4) Now, the diagonal formed basically represents the resultant vector in both magnitude and direction.

What are the Essential Conditions for the Addition of Vectors?

The essential condition for the addition of two vectors is simply that they should have the same dimensions and the same units. For example, a force vector with another force vector can be added, when they are expressed in the same units, but you cannot add force and velocity as they have different dimensions.

For example: If we have velocities of 30 meters/second and 50 meters/second in given directions we can add them easily but we can not directly add the velocities of say 3km/Second and 500 meters/second unless both are converted to the same units.

If the two vectors belong to the same vector space, they have the same dimension but it is also possible to add two vectors with different dimensions. For example, we have a vector A=3i+4j and a vector B=8i+5j+9k then we can also find a sum although they have different dimensions. Here we have to consider A=3i+4j+0k. The sum of the vectors A+B = 11i+9j+9k. In simple words, we can say that two vectors can be added if and only if they have the same unit.

Where Can We Use the Concept of the Addition of Vectors?

There are many fields where the concept of the addition of vectors can be used such as different fields of engineering like forces, magnetic fields, electric fields, momentum, position, trajectories, angular momentum, polarization, magnetization, kinetic density, torque, and velocities. Since laws of addition of vectors are fundamental mathematical laws, therefore, they are true and accepted for all vectors including vector quantities from fields of physics that are employed in engineering. 

Solved Questions 

1) Given the vectors A = 2i + 6j - 3k and B = 3i - 3j + 2k. Find A+B.

Ans. Let’s add the given vectors,

 A = 2i + 6j - 3k + B = 3i - 3j + 2k 

= (2+3)i + (6-3)j + (-3+2)k

Therefore, A+B = 5i+ 3j-1k

2) Predict the addition of vectors PQ and QR if PQ = (3, 2) and QR = (2, 6).

Ans. According to the question, PQ + QR = (3, 4) + (2, 6) which will be equal to (3 + 2, 4 + 6). Therefore, the value of PQ + QR will be (5, 10).

3) Calculate a + 2b - 3c if the position vectors a, b and c are given as A (3, 4), B (5, -6) and C (4, -1)?

Ans. Since, A, B and C are position vectors of the points A (3, 4), B (5, -6) and C (4, -1), therefore the corresponding vectors will be,

a = 3i + 4j

b = 5i - 6j

c = 4i - j

Now substitute these values of a, b, and c in a + 2b - 3c to calculate its value. On calculation, this value will come out as i - 5j.

4) The A, B, and C vertices of a triangle ABC have position vectors as a, b and c. Find the values of vectors AB + BC + CA.

Ans. According to the question, a, b and c represent the position vectors of vertices A, B, and C, therefore, in that case

Vector AB = b - a

Vector BC = c - b

Vector CA = a - c

Now, to calculate the value of AB + BC + CA, substitute the above values in the given formula. On calculation, the value of  AB + BC + CA will come out to be 0.

Questions for Self-Assessment and Practice

Here are some questions that are given for you to practice and evaluate your study of the concepts accordingly.

1) What will be the magnitude of the sum of displacement of 15 km and 25 km if the angle formed between them is 60 degrees?

2) Calculate the magnitude of the vector resultant from two vectors given as (2, 3) and (2, -2). Also, find the angle between the two vectors.

3) If the side BC of a triangle ABC has a D mid-point such that the sum of vectors AB + AC is equal to vector AD, then calculate the value of a.

4) Prove that the sum of three vectors determined using the median of a triangle and directed from the vertices is zero.