Addition of Vectors

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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

  1. Lay out the first vector besides a set of coordinate axes.

  2. Draw the first vector with its tail (base) at the point of inception of the coordinate axes.

  3. Locate the tail of the next vector over the head of the first one.

  4. Outlay a new vector from the inception point to the head of the last vector.

  5. 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.

FAQ (Frequently Asked Questions)

Q1. What is a Vector?

Vectors are typically the geometric renditions of magnitude (how far or how fast) and direction that can be described as arrows in 2 or 3 dimensions, commencing from one point on a coordinate axis and terminating at a different point. Vectors need both a magnitude and a direction. The magnitude of a vector is a number that helps to set a comparison of one vector to another. It also describes that the larger the magnitude, the longer the arrow. To describe a direction, there must be something relative to it, as the set of coordinates along the axis as shown below.

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Q2. What are the Properties of Vector Addition?

The Law of Vector addition adheres to the following properties.

  1. Vector addition is associative: While adding up 3 or more vectors together, the correlative grouping of vectors does not impact the result of collective addition.

Let 3 vectors→ P and Q and R be added together, then → P + (Q + R) = → (P+Q) + R

  1. Vector addition is commutative: by this principle, the order of vectors to be added does not impact the result of the addition. Let two vectors→ P and Q be added together, then → P + Q = → Q + P

  2. Vector addition is distributive: It suggests the sum of two vectors equals the sum of the scalar times of the two vectors, singularly, then

→ mP + mQ = m→ (P+Q)