Triangle Law of Vector Addition

Triangle Law of Vector Addition

State Triangle law of vector addition or State and explain triangle law of vector addition:

The triangle law of vector addition says that when two vectors are represented as two sides of a triangle with the same order of magnitude and direction, then the magnitude and direction of the resultant vector is represented by the third side of the triangle taken in reverse order.

Polygon law of vector addition

State polygon law of vector addition

The Polygon law of vector addition states that if the sides of a polygon are taken in the same order to represent a number of vectors in magnitude and direction, then the resultant vector can be represented in magnitude and direction by the closing side of the polygon taken in the opposite order.

(Source: Using Paint)

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. 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. Vector addition 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 vectors addition and these are:

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.

Also read -

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Triangle law of vector addition class 11

What is the Triangle law of forces of triangle 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.

(Source: Using Paint)

State and Prove Triangle law of Vector Addition or 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.

(Source: using ms tools)

We have,

R=P+Q

Expand A to C and draw perpendicular BC.

From triangle OCB we have,

OB2=OC2+BC2

OB2=(OA+OC)2+BC2 ….(i)

In the triangle ACB,

cosθ=AC/AB

AC=ABcosθ=Qcosθ

Also,

Sinθ=BC/AB

BC=ABsinθ=Qsinθ

Resultant magnitude:

Substitute the values for AC and BC in (i),

R2=(P+Qcosθ)2+(Qsinθ)2

R2=P2+2PQcosθ+Q2

R2=√(P2+2PQcosθ+Q2)

The above equation gives us the magnitude of the resultant.

Find the magnitude and direction of the resultant vector

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

From triangle OBC,

tan∅=BC/OC

Since, OC=OA+AC.

tan∅=BC/(OA+AC)

tan∅=Qsinθ/(P+Qcosθ)

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

Triangle law of vector addition examples

Example: Two vectors A and B of magnitude 5 units and 7 units respectively make an angle of 60o. 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:

R=|R|=√72+52+2*5*7cos60o

R=√25+49+70/2

R=√109units

Important Notes 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