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Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

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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) Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET (ii) Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET (iii) Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

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.

Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET
Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET [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.

Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

(i) Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET (ii) Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET (iii) Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEETAddition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

Note: and Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET thus Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET [Commutative Law]

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

Suppose angle between these two vectors is θ, and Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

(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 = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

or Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

angle α with vector a is

tan α = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

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.

  •  Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET and angle between them θ then R = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET
  •  Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET and angle between them π -θ then R = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET
  •  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, Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET
  •  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 Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET is a vector of the same magnitude as vector but pointing in a direction opposite to that of Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET.

Thus, Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET can be written as Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET or Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET is really the vector addition of Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET and Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET.

Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

Suppose angle between two vectors and is θ. Then angle between and Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET will be 180° -θ as shown in figure.

Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

Magnitude of Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET will be thus given by

S = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

or S = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET ...(i)

For direction of Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET we will either calculate angle α or β, where,

tanα = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET ...(ii)

or tanβ = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET ...(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. Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET ⇒ α = 30°

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

Note: or Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEETcan also be found by making triangles as shown in the figure. (a) and (b)

Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET                  Or                         Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

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

Sol. Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

Ex.5     Find Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET and Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET in the diagram shown in figure. Given A = 4 units and B = 3 units.

Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

Sol. Addition : 

R = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

= Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET units

tanα = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET = 0.472

Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEETa = tan-1(0.472) = 25.3°

Thus, resultant of Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET and Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET is Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET units at angle 25.3° from Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET in the direction shown in figure.

Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

Subtraction : S = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

= Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

and tanθ = Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

= Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET = 1.04

∴ α = tan-1 (1.04) = 46.1°

Thus, Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET is Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET units at 46.1° from Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET 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 Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET can be written as the product of unit vector Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET in that direction and magnitude of the given vector.

Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET or Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

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 Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET and respectively such that Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

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 Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET. A force of 105 dynes acts west wards. Represent the force in terms of Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET 

Sol.Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET

The document Addition & Subtraction of Vectors (Triangular & Parallelogram Laws) - Graphical Method - Notes | Study Physics Class 11 - NEET is a part of the NEET Course Physics Class 11.
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