Triangle Law of Vector Addition - Examples, FAQs

The Triangle Law of Vector Addition is a fundamental concept in physics and mathematics that helps us understand how to add two vectors geometrically. According to this law, if two vectors are represented as the two sides of a triangle, taken in order, their resultant is represented by the third side of the triangle, taken in the reverse order. This principle is widely used in fields like mechanics, navigation, and electromagnetism to analyze forces, velocities, and other vector quantities. In this article, we’ll break down the law, its derivation, and practical applications.

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This Story also Contains

1. Introduction to Vector Addition
2. Triangle Law of Vector Addition
3. Derivation of the Triangle Law of Vector Addition
4. Parallelogram Law of Vector Addition
5. Triangle Law of Forces
6. How to Find the Direction of the Resultant Vector?
7. Example of Triangle Law of Vector Addition
8. Important Points About Vector Addition
9. Summary

Introduction to Vector Addition

Vector Addition is the process of combining two or more vectors to determine a resultant vector, which represents the overall effect of those vectors. Vectors are quantities that have both magnitude and direction, such as force, velocity, or displacement. Unlike scalar quantities, which can be simply added arithmetically, vectors require both magnitude and direction to be considered in their addition.

There are two primary methods of vector addition: the Triangle Law and the Parallelogram Law. In both cases, the individual vectors are placed tail to head, and the resultant vector is drawn from the starting point to the endpoint.

Laws of Vector Addition

Vectors are written/represented with an alphabet and an arrow over them and are represented as a combination of direction and magnitude. The addition of two or more vectors is referred to as vector addition. When we add vectors, we use the addition operation to add two or more vectors to obtain a new vector that equals the sum of the two or more vectors. The addition of vector can be used to combine two vectors, a and b, and thus the resultant vector can be expressed as:

R = a+ b here bold is used to show that R, a and b are vectors

There are different laws of vector addition and these are:

Triangle Law of Vector Addition

The triangle law of vector addition states that if two vectors are represented by the sides of a triangle taken in order of magnitude and direction, then the resultant sum of the vectors is given by the triangle's third side in reverse order of magnitude and direction.

Mathematically, if you have two vectors A and B, and you place the tail of vector B at the head of vector A, then the resultant vector R is the vector drawn from the tail of A to the head of B. The direction and magnitude of this resultant vector represent the combined effect of both vectors.

Derivation of the Triangle Law of Vector Addition

Consider two vectors P and Q, represented in both magnitude and direction by the sides OA and AB of a triangle OAB, respectively. Let R be the product/resultant of the triangle law of vector additions. The resultant of P and Q is therefore represented by side OB according to the triangle law of addition or triangle law of vector addition.

We have,
$R=P+Q$
Expand A to C and draw perpendicular BC .
From triangle OCB we have,

$\begin{array}{rl}& {\mathrm{OB}}^2={\mathrm{OC}}^2+{\mathrm{BC}}^2\\ & {\mathrm{OB}}^2=(\mathrm{OA}+\mathrm{OC}{)}^2+{\mathrm{BC}}^2\end{array}$
In the triangle ACB,

$\begin{array}{rl}& \cos \theta =AC/AB\\ & AC=AB\cos \theta =Q\cos \theta \end{array}$
Also,
$\sin \theta =BC/AB$

$BC=AB\sin \theta =Q\sin \theta$
Resultant magnitude:
Substitute the values for $AC$ and $BC$ in (i),

$\begin{array}{rl}& R^2=(P+Q\cos \theta {)}^2+(Q\sin \theta {)}^2\\ & R^2=P^2+2PQ\cos \theta +Q^2\\ & R=\sqrt{}(P^2+2PQ\cos \theta +Q^2)\end{array}$

The above equation gives us the magnitude of the resultant.

Parallelogram Law of Vector Addition

The Parallelogram Law of Vector Addition is another fundamental method for determining the resultant of two vectors. This law states that if two vectors are represented as adjacent sides of a parallelogram, then the diagonal of the parallelogram that starts from the common point of the vectors represents their resultant.

Here's how it works:

1. Draw the Vectors: Start by placing the vectors A and B such that they both originate from the same point.
2. Form a Parallelogram: Complete the parallelogram by drawing lines parallel to A and B from the endpoints of each vector.
3. Draw the Resultant: The diagonal of the parallelogram that extends from the common point of the vectors to the opposite corner represents the resultant vector R.

Triangle Law of Forces

The Triangle Law of forces is applicable when there are three forces acting on a body in equilibrium. The two forces are then represented as two sides of a triangle in the same order, with their magnitude scaled to a suitable scale, and the resultant in the opposite order is the third side or closing side of the triangle.

The Triangle Law of Forces can also be used to calculate the resultant of two forces acting at a point.

Related Topic,

How to Find the Direction of the Resultant Vector?

The direction of the magnitude of the resultant vector is given by;

From triangle OBC,
$\tan \varnothing =\mathrm{BC}/\mathrm{OC}$
Since, $OC=OA+AC$.
$\tan \varnothing =BC/(OA+AC)$
$\tan \varnothing =\mathrm{Q}\sin \theta /(\mathrm{P}+\mathrm{Q}\cos \theta )$

The above equation gives us the direction of the resultant vector.

Recommended Topic Video

Example of Triangle Law of Vector Addition

Example: Two vectors A and B of magnitude 5 units and 7 units respectively make an angle of 60^o. Determine the magnitude of the resultant vector.

Solution: By following the triangle law of vector addition, the resultant vector is given by:
$R=A+B$
The magnitude of $R$ is:

$\begin{array}{rl}& R=|R|=\sqrt{}{7}^2+5^2+2^{\ast }{5}^{\ast }7\cos {60}^{\circ }\\ & R=\sqrt{}25+49+70/2\\ & R=\sqrt{}109\text{units}\end{array}$

Important Points About Vector Addition

Here is a list of some points to keep in mind while studying vector addition:

1. Vectors are depicted with an arrow and are represented as a combination of direction and magnitude.

2. If we know the components of a vector, we can calculate the direction of the resultant vector.

3. The well-known triangle law of forces can be used to add vectors, and this method is also known as the head-to-tail method.

Also, check-

• NCERT Exemplar Class 11th Physics Solutions
• NCERT Exemplar Class 12th Physics Solutions
• NCERT Exemplar Solutions for All Subjects

NCERT Physics Notes:

• NCERT Notes Class 11th Physics
• NCERT Notes Class 12th Physics
• NCERT Notes For All Subjects

Summary

The sum of two vectors can be found mathematically using the triangular law of vector addition. Mathematical physics includes vector addition and subtraction as essential concepts. A vector is an entity with both magnitude and direction, sometimes known as a quantity. It is a geometric addition of the vectors.