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Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET

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Mathematics and Science were invented by humans to understand and describe the world around us. A lot of mathematical quantities are used in Physics to explain the concepts clearly. A few examples of these include force, speed, velocity, and work. 

These quantities are often described as being a scalar or a vector quantity. 

  • Scalars and vectors are differentiated depending on their definition. A scalar quantity is defined as the physical quantity that has only magnitude, for example, mass and electric charge. 
  • On the other hand, a vector quantity is defined as the physical quantity that has both magnitudes as well as direction like force and weight. 
  • The other way of differentiating these two quantities is by using a notation. In this document, let us try to learn what is a vector and a scalar quantity.

What is a Scalar Quantity?

Scalar quantity is defined as the physical quantity with magnitude and no direction.

  • A scalar quantity only has a magnitude and a unit, it can be represented by a number only. A scalar does not have any direction.
    Example: Mass = 4 kg
  • The magnitude of mass = 4, Unit of mass = kg
  • Scalar quantities can be added, subtracted, and multiplied by simple laws of algebra.
  • Examples of Scalar Quantities: There are plenty of scalar quantity examples, some of the common examples are: Mass, Speed, Distance, Time, Area, Volume, Density & Temperature

Question for Vector & Scalar Quantities
Try yourself:Which of the following is a scalar quantity?
View Solution


What is a Vector Quantity?

A vector quantity is defined as the physical quantity that has both direction as well as magnitude.

  • Vector is the physical quantities having magnitude as well as specified direction.Specified Direction
    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 |A|.
  • Example of Vector quantity: Displacement, velocity, acceleration, force, etc.

Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET


Table: Difference between Vector and Scalar

Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET


General Points Regarding Vectors

1. Representation of Vector
  • Geometrically, the vector is represented by a line with an arrow indicating the direction of the vector as:

VectorVector

  • Mathematically, the vector is represented by Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET. Sometimes it is represented by the bold letter A.

Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET

  • Thus, the arrow in the above figure represents a vectorVector & Scalar Quantities - Notes | Study Physics Class 11 - NEET in XY-plane making an angle θ with the x-axis.
  • A representation of the vector will be complete if it gives us direction and magnitude.
  • Symbolic form: Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET 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.
    Example: If we want to represent a force of 5 N acting 45° N of EVector & Scalar Quantities - Notes | Study Physics Class 11 - NEET(i) We choose direction coordinates.
    (ii) We choose a convenient scale like 1 cm = 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 an arrow in the direction of the vector.
2. The Angle Between Two Vectors (θ)

The 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.

Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET

3.  Negative of Vector

It implies vectors of the same magnitude but opposite in direction.

Negative vectorNegative vector

4. Equality of Vectors

Vectors having equal magnitude and same direction are called equal vectors.

Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET

Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET

5. Collinear Vectors

Any two vectors are collinear, then one can be expressed in terms of others. Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET = Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET (where λ is a constant)

Collinear vectorCollinear vector

Question for Vector & Scalar Quantities
Try yourself:What is the magnitude of a unit vector?
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6. Co-initial Vector

If two or more vectors start from the same point, then they called co-initial vectors.

Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET

Co-initial vector

Here A, B, C, D are co-initial vectors.


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.

Coplanar VectorsCoplanar Vectors

Note:

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


8. Multiplication and Division of a Vector by a Scalar
  • Multiplying a vectorVector & Scalar Quantities - Notes | Study Physics Class 11 - NEET with a positive number λ gives a vectorVector & Scalar Quantities - Notes | Study Physics Class 11 - NEET whose magnitude becomes λ times, but the direction is the same as that ofVector & Scalar Quantities - Notes | Study Physics Class 11 - NEET.
    Multiplying a vectorVector & Scalar Quantities - Notes | Study Physics Class 11 - NEET by a negative number λ gives a vector Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET whose direction is opposite to the direction of  Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET and whose magnitude is λ times Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET.
  • The division of vector Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET by a non-zero scalar 'm' is defined as the multiplication of Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET by Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET
  • At here Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET and Vector & Scalar Quantities - Notes | Study Physics Class 11 - NEET are colinear vector.
9. Multiplication of a Vector
  • By a Real Number
    When a vector A is multiplied by a real number n, then its magnitude becomes n times but direction and unit remains unchanged.
  • By a Scalar
    When a vector A is multiplied by a scalar S, then its magnitude becomes S times, and unit is the product of units of A and S but direction remains same as that of vector A.
10. Scalar or Dot Product of Two Vectors
  • The scalar product of two vectors is equal to the product of their magnitudes and the cosine of the smaller angle between them. It is denoted by . (dot)
    A . B = AB cos θ
  • The scalar or dot product of two vectors is a scalar.

Properties of Scalar Product

  • Scalar product is commutative, i.e.,
    A . B= B . A
  • Scalar product is distributive, i.e.,
    A . (B + C) = A . B + A . C
  • Scalar product of two perpendicular vectors is zero.
    A . B = AB cos 90° = 0
  • Scalar product of two parallel vectors is equal to the product of their magnitudes, i.e., A . B = AB cos 0° = AB
  • Scalar product of a vector with itself is equal to the square of its magnitude, i.e.,
    A . A = AA cos 0° = A2
  • Scalar product of orthogonal unit vectors and Scalar product in cartesian coordinates
    = AxBx + AyBy + AzBz

Question for Vector & Scalar Quantities
Try yourself:Dot product of two mutual perpendicular vector is
View Solution


11. Vector or Cross Product of Two Vectors
  • The vector product of two vectors is equal to the product of their magnitudes and the sine of the smaller angle between them. It is denoted by X (cross).
    A X B = AB sin θ n
    Vector cross product
    Vector cross product
  • The direction of unit vector n can be obtained from right hand thumb rule.
  • If fingers of right hand are curled from A to B through smaller angle between them, then thumb will represent the direction of vector (A X B).
  • The vector or cross product of two vectors is also a vector.

Properties of Vector Product

  • Vector product is not commutative, i.e.,
    A X B ≠ B X A  [∴ (A X B) = - (B X A)]
  • Vector product is distributive, i.e.,
    A X (B + C) = A X B + A X C
  • Vector product of two parallel vectors is zero, i.e.,
    A X B = AB sin 0° = 0
  • Vector product of any vector with itself is zero.
    A X A = AA sin 0° = 0
  • Vector product of orthogonal unit vectors.
  • Vector product in cartesian coordinates.

Question for Vector & Scalar Quantities
Try yourself:The angle between the vectors (A x B) and (B x A) is:
View Solution


Direction of Vector Cross Product

  • When C = A X B, the direction of C is at right angles to the plane containing the vectors A and B. The direction is determined by the right hand screw rule and right hand thumb rule.
  • Right Hand Thumb Rule Curl the fingers of your right hand from A to B. Then, the direction of the erect thumb will point in the direction of A X B.
    Right Hand Thumb Rule
    Right Hand Thumb Rule
  • Right Hand Screw Rule Rotate a right handed screw from first vector (A) towards second vector (B). The direction in which the right handed screw moves gives the direction of vector (C).

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