What is a Vector?
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.
(Image to be added soon)
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.
(Image to be added soon)
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.
(Image to be added soon)
Now after this, we need to join the other endpoints of both the vectors together as shown below,
(Image to be added soon)
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.
(Image to be added soon)
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.
(Image to be added soon)
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.
FAQs on Laws of Vector Addition
1. What are the main laws for vector addition taught in Class 11?
The two fundamental laws for vector addition as per the CBSE curriculum are:
- Triangle Law of Vector Addition: Used for adding two vectors by placing them head-to-tail. The resultant vector is the third side of the triangle, closing the loop.
- Parallelogram Law of Vector Addition: Used when two vectors act from the same point. They form the adjacent sides of a parallelogram, and the diagonal from the common point represents the resultant vector.
A third, the Polygon Law, is an extension used for adding more than two vectors.
2. How does the Triangle Law of Vector Addition work with an example?
The Triangle Law states that if two vectors are represented by two sides of a triangle in sequence (head of the first vector connected to the tail of the second), then the third side of the triangle, drawn from the tail of the first vector to the head of the second, represents the resultant vector in both magnitude and direction. For example, if you walk 4 km East and then 3 km North, your net displacement is the hypotenuse of the triangle formed, which can be found by adding the two displacement vectors.
3. What is the Parallelogram Law of Vector Addition?
The Parallelogram Law of Vector Addition is used to find the resultant of two vectors originating from the same point. It states that if two vectors are represented by the two adjacent sides of a parallelogram drawn from a common point, their resultant sum is represented by the diagonal of the parallelogram that also passes through that same common point.
4. What are the most important properties of vector addition?
Vector addition follows specific algebraic properties that are crucial for solving problems:
- Commutative Law: The order of addition does not matter. For any two vectors A and B, A + B = B + A.
- Associative Law: When adding three or more vectors, the grouping does not affect the result. For any three vectors A, B, and C, (A + B) + C = A + (B + C).
5. Why can't we add vectors representing different physical quantities, like a force vector and a velocity vector?
We cannot add vectors of different physical quantities because of the principle of dimensional homogeneity. For an addition operation to be physically meaningful, the quantities being added must have the same units and dimensions. A force is measured in Newtons (kg·m/s²) while velocity is in meters per second (m/s). Adding them would result in a mathematically valid vector but a physically nonsensical quantity.
6. How is vector subtraction different from vector addition?
Vector subtraction is essentially a special case of vector addition. To subtract a vector B from a vector A, you add the negative of vector B to A. The negative of a vector (-B) is a vector with the same magnitude as B but pointing in the exact opposite direction. Therefore, the operation A - B is performed as A + (-B) using the standard laws of vector addition.
7. When is it better to use the Triangle Law versus the Parallelogram Law?
Both laws give the same resultant vector, but their application is conceptually different. The Triangle Law is more intuitive for sequential actions, such as calculating the total displacement after two separate journeys. The Parallelogram Law is generally more useful for situations where two forces or fields act simultaneously on a single point, like finding the net force on an object being pulled in two different directions at once.
8. What is the Polygon Law of Vector Addition and when is it necessary?
The Polygon Law of Vector Addition is an extension of the Triangle Law used for finding the resultant of more than two vectors. You arrange all the vectors head-to-tail in sequence, forming the open sides of a polygon. The resultant vector is the final closing side of the polygon, drawn from the tail of the very first vector to the head of the very last vector.
9. Can you provide a real-world example of the parallelogram law of vector addition?
A classic example is a boat crossing a river. The boat has a velocity vector pointing across the river, while the river current has its own velocity vector pointing downstream. Both forces act on the boat at the same time. The boat's actual path and speed relative to the riverbank is the resultant vector, which is the diagonal of the parallelogram formed by the boat's velocity vector and the river's current vector.
