Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes

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The document Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes is a part of the Class 11 Course Physics Class 11.
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1. Scalar

In physics we deal with two types of physical quantity one is scalar and other is vector. Each scalar quantity has a magnitude and a unit.

For example mass = 4 kg

Magnitude of mass = 4

and unit of mass = kg

Example of scalar quantities: mass, speed, distance etc.

Scalar quantities can be added, subtracted and multiplied by simple laws of algebra.

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev NotesFig: Scalars and vectors


2. Vector

Vector are the physical quantities having magnitude as well as specified direction.

For example:

Speed = 4 m/s (is a scalar)

Velocity = 4 m/s toward north (is a vector)

If someone wants to reach some location then it is not sufficient to provide information about the distance of that location it is also essential to tell him about the proper direction from the initial location to the destination.

The magnitude of a vector () is the absolute value of a vector and is indicated by Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes or A.

Example of vector quantity : Displacement, velocity, acceleration, force                                                                                            etc.

Knowledge of direction: 

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev NotesFig: Specified Direction

3. General Points Regarding Vectors

3.1 Representation of vector:

Geometrically, the vector is represented by a line with an arrow indicating the direction of vector as:

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes

Fig: Vector

Mathematically, vector is represented by Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes.

Sometimes it is represented by bold letter A.

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes

Thus, the arrow in the above figure represents a vector Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes in xy-plane making an angle θ with x-axis.

 A representation of vector will be complete if it gives us direction and magnitude.

Symbolic form: Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes used to separate a vector quantity from scalar quantities (u, i, m)

Graphical form: A vector is represented by a directed straight line,having the magnitude and direction of the quantity represented by it.

e.g. if we want to represent a force of 5 N acting 45° N of E

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes

(i) We choose direction coordinates.

(ii) We choose a convenient scale like 1 cm º  = 1 N

(iii) We draw a line of length equal in magnitude and in the direction of vector to the chosen quantity.

(iv) We put arrow in the direction of vector.

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes

Magnitude of vector:

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes


3.2 Angle between two Vectors (θ):

Angle between two vectors means smaller of the two angles between the vectors when they are placed tail to tail by displacing either of the vectors parallel to itself (i.e 0 £ q £ p).

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes

 

Ex.1 Three vectors Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes are shown in the figure. Find angle between (i) Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes and Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes, (ii) Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes and Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes, (iii) Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes and Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev NotesTypes of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev NotesTypes of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes

Ans: To find the angle between two vectors we connect the tails of the two vectors. We can shift Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes & Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notessuch that tails of Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes and Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes are connected as shown in figure.

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes

Now we can easily observe that angle between Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes and Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes is 60º, Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes and Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes is 15º and between Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes and Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes is 75º.


3.3 Negative of Vector: 

It implies vector of same magnitude but opposite in direction.

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes


3.4 Equality of Vectors: 

Vectors having equal magnitude and same direction are called equal vectors

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes


3.5 Collinear vectors: 

Any two vectors are collinear then one can be expressed in the terms of others.

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes = Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes (where l is a constant)

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes


3.6 Co-initial vector: If two or more vector start from the same point then they called co-initial vector.

e.g.

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes

here A, B, C, D are co-initial.


3.7 Coplanar vectors: 

Three (or more) vectors are called coplanar vectors if they lie in the same plane or are parallel to the same plane. Two (free) vectors are always coplanar.

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes

Important points:

  • A If the frame of reference is translated or rotated the vector does not change (though its components may change).

Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes

Two vectors are called equal if their magnitudes and directions are same, and they represent values of same physical quantity.


3.8 Multiplication and division of a vector by a scalar:

Multiplying a vector Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes with a positive number λ gives a vector Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes whose magnitude become λ times but the direction is the same as that of Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes. Multiplying a vector Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes by a negative number λ gives a vector Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes whose direction is opposite to the direction of  Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes and whose magnitude is -λ times Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes.

The division of vector Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes by a non-zero scalar m is defined as multiplication of Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes by Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes

At here Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes and Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes are colinear vector


Ex.2 A physical quantity (m = 3 kg) is multiplied by a vector Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes such that Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes. Find the magnitude and direction of Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes if 

(i) Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes = 3m/s2 East wards 

(ii) Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes = -4 m/s2 North wards 

Ans: (i) Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes East wards

= 9 N East wards

(ii) Types of Vectors and Scalars - Motion in a Plane, Class 11, Physics | EduRev Notes North wards

= -12 N North wards

= 12 N South wards

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