Vector Law of Addition
One of the techniques in which describing physical quantities as vectors makes evaluations much easier is the ease with which vectors can be added to one another. Keeping in consideration that the vectors are graphical representation, addition and subtraction of vectors can be performed graphically. For vector addition, one need not bother about which vector to draw first seeing that addition is commutative. However, for subtraction, you need to make sure that the vector you draw first is the vector you are subtracting from.
Graphical Representation of a Vector Addition
The graphical method of vector addition is also termed as the “head-to-tail method”. To begin with, you need to
Lay out the first vector besides a set of coordinate axes.
Draw the first vector with its tail (base) at the point of inception of the coordinate axes.
Locate the tail of the next vector over the head of the first one.
Outlay a new vector from the inception point to the head of the last vector.
This resultant vector (new line) is the sum of the original two.
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Laws of Vector Addition with Examples
Physical theories such as acceleration and velocity are all examples of quantities that can be described by vectors. Each of these quantities has both a magnitude and a direction. An example of vector addition in physics is as below:-
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Laws of Vector Addition
The addition of two vectors may be easily understood by the following laws i.e.
law of triangle
Law of a parallelogram
Triangle’s Law of Vector Addition
Triangle law of addition states the addition of two vectors which can be described as follows:
“If 2 vectors are illustrated (in magnitude and direction) by the two sides of a triangle, taken in the similar order, then their consequential vector is represented (in magnitude and direction) by the third side of the triangle is taken in the opposite order.”
Triangle’s Law of Vector Addition with Example
Let 2 vectors be →A and →B acting at the same time on the plane.
Now, delineate Vector→ B by the line [OB]. At point A, draw another line →OA, speaking of the vector → [B]. Time is to connect OC. Then the vector → [OC] is equal to →R that provides the results of the vector →[A] and →[B].
It can be noted that vectors →[OB] and →[BC] come about in the same order while →[R] is in the opposite order. Thus, validates compliance with the triangle’s law.
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Parallelogram of Vector Addition
When two vectors acting concurrently on a particle are represented in magnitude and direction by the two adjoining sides of a parallelogram drawn from a point, then their resultant is entirely presented in magnitude and direction respectively by the diagonal of that parallelogram drawn from that point.
Take into account 2 vectors →P and →Q behaving simultaneously on a particle O at an angle 0. They are representative of magnitude and direction by the adjoining sides OA and OB of a parallelogram “OACB” outlaid from a point O. At that time, the diagonal ‘OC’ crosses through ‘O’, and will constitute the resultant R in magnitude and direction.
That said, If Q is in displacement from position OB to AC by displacing, this method gets in correspondence to the triangle method.
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Using Components to Add Vectors
Another potential way to add vectors is by adding the components. Earlier, we observed that vectors can be represented in terms of their vertical and horizontal components. For the purpose to add vectors, solely represent both in respect to their horizontal and vertical components and then add up the components altogether.
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Solved Examples
Example1:
A vector ‘P’ with a magnitude of 5 at an angle of 36.9° to the horizontal axis will have a vertical component of 3 units and a horizontal component of 4.
Solution1:
On the assumption to add this to another vector of the same magnitude and direction, we would get a vector twice as long at the same angle.
Observe this by adding the horizontal components of both the vectors
With which you get,
4 + 4 (two horizontal components)
And
3 + 3 (two vertical components)
With the additions, we now obtain
A new vector including a horizontal component of 8 (4+4)…… And
A vertical component of 6 (3+3)
Thus, to determine the resultant vector
Just locate the tail of the vertical component (at the head) of the horizontal component and then pull out a line from the point of inception (to the head) of the vertical component.
Hence, we get a new line as the resultant vector
Take note that, it must be twice as long as the original, being so that both of its components are two times as large as they were previously.
Fun Facts
The angle made horizontally can be used to compute the magnitude of the two components.
Vectors are actually physical quantities that are applied to not just mathematics but in physics and engineering.
FAQs on Addition of Vectors
1. What is a vector and how is it represented graphically in Maths?
A vector is a mathematical object that has both magnitude (size or length) and direction. Unlike scalars, which only have magnitude (like speed or temperature), vectors provide directional information. Graphically, a vector is represented as a directed line segment or an arrow. The length of the arrow corresponds to the vector's magnitude, and the arrowhead points in its specific direction.
2. What are the two main graphical methods for adding vectors as per the CBSE syllabus?
The two primary graphical methods for vector addition are:
- Triangle Law of Vector Addition: If two vectors are represented by two sides of a triangle in sequence (head to tail), then the third side of the triangle, drawn from the starting point of the first vector to the ending point of the second, represents the resultant vector in both magnitude and direction.
- Parallelogram Law of Vector Addition: If two vectors are represented by the two adjacent sides of a parallelogram drawn from a common point, their sum (or resultant vector) is represented by the diagonal of the parallelogram that passes through that same common point.
3. How do you find the magnitude and direction of the resultant vector when adding two vectors?
To find the sum, or resultant (R), of two vectors A and B, you can use the analytical method. The magnitude of the resultant vector is calculated using the formula derived from the Law of Cosines: |R| = √(A² + B² + 2ABcosθ), where θ is the angle between vectors A and B. The direction of the resultant vector can be found by calculating the angle (α) it makes with one of the original vectors, typically using the Law of Sines.
4. What are the fundamental properties of vector addition?
Vector addition follows several key properties that are similar to scalar addition, but applied to vector quantities:
- 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).
- Existence of Additive Identity: There exists a zero vector (or null vector), denoted as 0, which when added to any vector, leaves the vector unchanged (a + 0 = a).
5. What is the key difference between adding vectors and adding scalars?
The fundamental difference is the consideration of direction. When you add scalars (like 5 kg + 10 kg), you simply add their magnitudes to get 15 kg. However, when you add vectors, you must account for both their magnitudes and their directions. For example, adding two force vectors of 5N and 10N could result in a total force anywhere from 5N (if they oppose each other) to 15N (if they act in the same direction), depending on the angle between them.
6. Can any two vectors be added together? What is the necessary condition?
No, not any two vectors can be meaningfully added. The essential condition for vector addition is that the vectors must represent the same physical quantity. For instance, you can add a velocity vector to another velocity vector, or a force vector to another force vector. However, it is physically meaningless to add a velocity vector (measuring motion) to a force vector (measuring a push or pull).
7. How is vector addition applied in a real-world scenario, like finding a boat's path across a river?
Vector addition is crucial for solving problems in physics and engineering. For example, consider a boat trying to cross a river. The boat has its own velocity vector (pointing across the river), and the river has a current velocity vector (pointing downstream). The actual path and speed of the boat relative to the ground is the resultant vector, found by adding the boat's velocity vector and the river's current vector using the Triangle or Parallelogram Law. This helps predict where the boat will actually land on the opposite bank.
8. What is the result of adding a vector to its negative counterpart?
Adding a vector to its negative counterpart results in a zero vector (or null vector). A negative vector has the same magnitude as the original vector but points in the exact opposite direction (an angle of 180°). When you add them, for example using the Triangle Law, the 'head' of the second vector ends up at the 'tail' of the first vector, resulting in a vector with zero magnitude and no specific direction. This is conceptually similar to how 5 + (-5) = 0 in scalar arithmetic.
