Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

Physics Class 11

NEET : Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

The document Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev is a part of the NEET Course Physics Class 11.
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4. LAWS OF ADDITION AND SUBTRACTION OF VECTORS

4.1 Triangle rule of addition: Steps for addicting two vector representing same physical quantity by triangle law.

(i) Keep vectors s.t. tail of one vector coincides with head of others.

(ii) Join tail of first to head of the other by a line with arrow at head of the second.

(iii) This new vector is the sum of two vectors. (also called resultants)

(i) Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev (ii) Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev (iii) Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

Take example here.

Q. A boy moves 4 m south and then 5 m in direction 37° E of N. Find resultant displacement.

4.2 Polygon Law of addition : 

This law is used for adding more than two vectors. This is extension of triangle law of addition. We keep on arranging vectors s.t. tail of next vector lies on head of former.

When we connect the tail of first vector to head off last we get resultant of all the vectors.

Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev
Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev [Associative Law]

4.3 Parallelogram law of addition : 

Steps : 

(i) Keep two vectors such that their tails coincide.

(ii) Draw parallel vectors to both of them considering both of them as sides of a parallelogram.

(iii) Then the diagonal drawn from the point where tails coincide represents the sum of two vectors, with its tail at point of coincidence of the two vectors.

Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

(i) Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev (ii) Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev (iii) Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRevTriangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

Note: and Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev thus Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev [Commutative Law]

Note: Angle between 2 vectors is the angle between their positive directions.

Suppose angle between these two vectors is θ, and Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

(AD)2 = (AE)2 +(DE)2

= (AB + BE)2 + (DE)2

= (a +b cosθ)2 + (b sinθ)

= a2 + b2 cos2θ + 2ab cosθ + b2 sin2θ

= a2 + b2 + 2ab cosθ

Thus, AD = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

or Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

angle α with vector a is

tan α = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

Important points :

  •  To a vector, only a vector of same type can be added that represents the same physical quantity and the resultant is also a vector of the same type.
  •  As R = [A2 + B2 + 2AB cosθ]1/2 so R will be maximum when, cosθ = max = 1,

               i.e., θ = 0º, i.e. vectors are like or parallel and Rmax = A + B.

  •  Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev and angle between them θ then R = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev
  •  Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev and angle between them π -θ then R = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev
  •  The resultant will be minimum if, cosθ = min = -1, i.e., θ = 180º, i.e. vectors are antiparallel and Rmin = A -B.
  •  If the vectors A and B are orthogonal, i.e., θ = 90º, Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev
  •  As previously mentioned that the resultant of two vectors can have any value from (A -B) to (A + B)      depending on the angle between them and the magnitude of resultant decreases as q increases 0º to 180º.
  •  Minimum number of unequal coplanar vectors whose sum can be zero is three.
  •  The resultant of three non-coplanar vectors can never be zero, or minimum number of non coplanar vectors whose sum can be zero is four.

 5. SUBTRACTION OF VECTORS :

Negative of a vector say Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev is a vector of the same magnitude as vector but pointing in a direction opposite to that of Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev.

Thus, Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev can be written as Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev or Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev is really the vector addition of Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev and Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev.

Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

Suppose angle between two vectors and is θ. Then angle between and Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev will be 180° -θ as shown in figure.

Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

Magnitude of Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev will be thus given by

S = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

or S = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev ...(i)

For direction of Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev we will either calculate angle α or β, where,

tanα = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev ...(ii)

or tanβ = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev ...(iii)

Ex.3 Two vectors of 10 units & 5 units make an angle of 120° with each other. Find the magnitude & angle of resultant with vector of 10 unit magnitude. 

Sol. Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev ⇒ α = 30°

[Here shows what is angle between both vectors = 120° and not 60°]

Note: or Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRevcan also be found by making triangles as shown in the figure. (a) and (b)

Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev                  Or                         Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

Ex.4 Two vectors of equal magnitude 2 are at an angle of 60° to each other find magnitude of their sum & difference. 

Sol. Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

Ex.5     Find Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev and Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev in the diagram shown in figure. Given A = 4 units and B = 3 units.

Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

Sol. Addition : 

R = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

= Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev units

tanα = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev = 0.472

Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduReva = tan-1(0.472) = 25.3°

Thus, resultant of Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev and Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev is Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev units at angle 25.3° from Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev in the direction shown in figure.

Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

Subtraction : S = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

= Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

and tanθ = Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

= Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev = 1.04

∴ α = tan-1 (1.04) = 46.1°

Thus, Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev is Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev units at 46.1° from Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev in the direction shown in figure.

6. Unit Vector and Zero vector

Unit vector is a vector which has a unit magnitude and points in a particular direction. Any vector Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev can be written as the product of unit vector Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev in that direction and magnitude of the given vector.

Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev or Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

A unit vector has no dimensions and unit. Unit vectors along the positive x-, y-and z-axes of a rectangular coordinate system are denoted by Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev and respectively such that Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

A vector of zero magnitudes is called zero or a null vector. Its direction is arbitrary.

Ex.6 A unit vector along East is defined as Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev. A force of 105 dynes acts west wards. Represent the force in terms of Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev 

Sol.Triangular and Parallelogram Laws of Addition and Subtraction of Vectors Class 11 Notes | EduRev

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