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Triangle law of vector addition

The triangle law does not restrict where to start i.e. with which vector to start. Also, it does not put conditions with regard to any specific direction for the sequence of vectors, like clockwise or anti-clockwise, to be maintained. In figure (i), the law is applied starting with vector, b ; whereas the law is applied starting with vector, a , in figure (ii). In either case, the resultant vector, c , is same in magnitude and direction.

This is an important result as it conveys that vector addition is commutative in nature i.e. the process of vector addition is independent of the order of addition. This characteristic of vector addition is known as “commutative” property of vector addition and is expressed mathematically as :

a + b = b + a

If three vectors are represented by three sides of a triangle in sequence, then resultant vector is zero. In order to prove this, let us consider any two vectors in sequence like AB and BC as shown in the figure. According to triangle law of vector addition, the resultant vector is represented by the third closing side in the opposite direction. It means that :

Three vectors

Three vectors are represented by three sides in sequence.

AB + BC = AC

Adding vector CA on either sides of the equation,

AB + BC + CA = AC + CA

The right hand side of the equation is vector sum of two equal and opposite vectors, which evaluates to zero. Hence,

Three vectors

The resultant of three vectors represented by three sides is zero.

AB + BC + CA = 0

Note : If the vectors represented by the sides of a triangle are force vectors, then resultant force is zero. It means that three forces represented by the sides of a triangle in a sequence is a balanced force system.

Parallelogram law

Parallelogram law, like triangle law, is applicable to two vectors.

Parallelogram law
If two vectors are represented by two adjacent sides of a parallelogram, then the diagonal of parallelogram through the common point represents the sum of the two vectors in both magnitude and direction.

Parallelogram law, as a matter of fact, is an alternate statement of triangle law of vector addition. A graphic representation of the parallelogram law and its interpretation in terms of the triangle is shown in the figure :

Parallelogram law

Converting parallelogram sketch to that of triangle law requires shifting vector, b , from the position OB to position AC laterally as shown, while maintaining magnitude and direction.

Polygon law

The polygon law is an extension of earlier two laws of vector addition. It is successive application of triangle law to more than two vectors. A pair of vectors ( a , b ) is added in accordance with triangle law. The intermediate resultant vector ( a + b ) is then added to third vector (c) again, successively till all vectors to be added have been exhausted.

Successive application of triangle law

Polygon law
Polygon law of vector addition : If (n-1) numbers of vectors are represented by (n-1) sides of a polygon in sequence, then n th side, closing the polygon in the opposite direction, represents the sum of the vectors in both magnitude and direction.

Questions & Answers

if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
Abhyanshu Reply
x=5.8-3.22 x=2.58
sajjad
what is the definition of resolution of forces
Atinuke Reply
what is energy?
James Reply
can anyone tell who founded equations of motion !?
Ztechy Reply
n=a+b/T² find the linear express
Donsmart Reply
أوك
عباس
Quiklyyy
Sultan Reply
Moment of inertia of a bar in terms of perpendicular axis theorem
Sultan Reply
How should i know when to add/subtract the velocities and when to use the Pythagoras theorem?
Yara Reply
Centre of mass of two uniform rods of same length but made of different materials and kept at L-shape meeting point is origin of coordinate
Rama Reply
A balloon is released from the ground which rises vertically up with acceleration 1.4m/sec^2.a ball is released from the balloon 20 second after the balloon has left the ground. The maximum height reached by the ball from the ground is
Lucky Reply
work done by frictional force formula
Sudeer Reply
Torque
Misthu Reply
Why are we takingspherical surface area in case of solid sphere
Saswat Reply
In all situatuons, what can I generalize?
Cart Reply
the body travels the distance of d=( 14+- 0.2)m in t=( 4.0 +- 0.3) s calculate it's velocity with error limit find Percentage error
Clinton Reply

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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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