Unit – I A

Introduction: Introduction to scalars and vectors

Scalars:

  • The physical quantities which have magnitude only and which can be specified by a number and unit only are called as scalar quantities or scalars.
  • For e.g. when we are specifying time we may say like 20 seconds, 1 year, 24 hours etc. Here we are giving magnitude only i.e. a number and a unit. In this case, a direction is not required.
  • Examples of Scalars: Time, distance, speed, mass, density, area, volume, work, pressure, energy etc.

    Characteristics of Scalars:

  1. The scalar quantities have magnitude only.
  2. The scalars can be added or subtracted from each other algebraically.
  3. When writing scalar quantity an arrow is not put on the head of the symbol of the quantity.
  4. Mathematical operations like Addition, Subtraction, Multiplication and Division are valid in scalar algebra.

Vectors:

  • The physical quantities which have both the magnitude as well as the direction and which should be specified by both magnitude and direction are called as vector quantities or vectors.
  • For e.g. when we are specifying the displacement of the body, we have to specify the magnitude and direction. Hence, displacement is a vector quantity.
  • Examples of vectors: Displacement, velocity, acceleration, force, momentum, electric intensity, etc.

    Characteristics of Vectors:

  1. The vector quantities have both a magnitude and direction.
  2. The vectors can not be added or subtracted from each other algebraically but we have to adopt the graphical method.
  3. When writing vector quantity an arrow is put on the head of the symbol of the quantity.
  4. Mathematical operations like Addition, Subtraction and Multiplication are valid in vector algebra. Division of one vector by another vector is not a valid operation in vector algebra.

Representation of a Vector:

  • Vector is represented by a directed line segment AB. The length of segment AB is the magnitude of the vector and its direction is from A to B along the line AB.
  • The point A is called the initial point or tail of the vector and the point B is called the terminal point or head of a vector. The line AB along which the vector acts is called line of action of vector

  • Its magnitude is written as AB.
  •  The two end points A and B are not interchangeable.
  • The length of a directed line segment  is the distance between point A and point B
  • The line of an unlimited length of which a directed line segment is a part is called the Support of the vector.
    The Sense of the directed line segment is from its initial point to its terminal point.

Important Definitions in Vector Algebra:

Negative Vector:

  • Negative vector is the vector which has the same magnitude as that of given vector but has opposite direction to that of given vector.

Negative vector

Equal Vectors:

  • Two vectors are said to be equal if they have the same magnitude and the same direction.

Equal Vectors

  • If a vector is moved parallel to itself, it represents a vector equal to itself i.e. the same vector.

Unit Vector:

  • The vector having unit magnitude is called unit vector.

Unit vector magnitude



Unit vector

  • A vector can be written as magnitude times the unit vector along its own direction.

Co-initial Vectors:

  • Two vectors are said to be co-initial if they have a common initial point.

Co-initial vectors

Collinear Vectors:

  • Two vectors having equal or unequal magnitudes, which either act along the same line or along the parallel lines in the same direction or along the parallel lines opposite direction, are called collinear vectors.

Collnear vectors

Null Vector:

  • A vector having zero magnitude is called as null vector or zero vector.
  • For null Vector, initial and the terminal points coincide.
  • Any non-zero vector is called a proper vector.

Free Vector:

  • When there is no restriction to choose the origin of the vector, it is called as a free vector.

Localised Vector:

  • When there is a restriction to choose the origin of the vector, it is called as a localised vector.

Like Vectors:

  • Vectors having the same direction are called like vectors.

Unlike Vectors:

  • Vectors having opposite directions are called, unlike vectors.

Coplanar Vectors:

  • Vectors are said to be coplanar if they lie in the same plane or parallel to one and the same plane.