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When a physical quantity is measured, the measured value is always a number and this number will make sense only when the relevant unit is specified. Therefore, the result of a measurement has a numerical value and a unit of measure. The numerical value together with the unit is called the magnitude.

Physical quantities that can be described completely using only the magnitude, i.e., the numerical value and its unit are called scalar quantities, e.g. distance, work, volume, etc. If you say that the volume of water in a beaker is 25 cm3, it completely describes the physical quantity.

Physical quantities that need both magnitude and direction for their complete description are called vector quantities, e.g. displacement, force, velocity, etc. If we say that a body is moving at 5 m s−1, the description is incomplete. It also requires to be told as to in what direction it is moving.

Representation of a Vector

A vector quantity is associated with both direction and magnitude. Therefore, it is convenient to represent a vector graphically.

To represent a vector graphically
  1. draw a line whose length is proportional to the magnitude of the vector by choosing a suitable scale and
  2. indicate the direction of the vector by marking or pointing an arrow head at one end of the line as shown in Figure 1.2.
Description: 36061.png represents the velocity of the car with the arrow head showing the direction and the length representing the magnitude of velocity. The magnitude of the vector Description: 36069.png is denoted by Description: 36081.png called the modulus of the vector and is always positive.


Position Vector

To locate the position of any point P in a plane or space, generally a fixed point of reference called the origin O is taken. The vector is called the position vector of P with respect to O (Figure 1.3).
  • Given a point P, there is one and only one position vector for the point with respect to the given origin O.
  • The position vector of origin is a zero vector.
  • Position vector of a point P changes if the position of the origin O is changed.


Properties of Vectors

Vector Addition Two or more vectors can be added only if they have same units.
For example, we cannot add a velocity vector to a force vector as they correspond to different physical quantities.

It is very convenient to add two vectors using a graphical procedure. There are two methods of vector addition:
  1. triangle method of vector addition and
  2. parallelogram method of vector addition.
Triangle Method of Vector Addition Consider two vectors Description: 36113.png and Description: 36124.png representing the same physical quantity.
To add a vector Description: 42267.png to vector Description: 42249.png, first draw vector Description: 42254.png, with its magnitude represented by a convenient scale, and then draw vector Description: 42276.png to the same scale with its tail starting from the tip of Description: 42258.png as shown in Figure 1.4. The resultant vector Description: 36167.png is the vector drawn from the tail of Description: 42262.png to the tip of Description: 42281.png.


Parallelogram Law of Vector Addition In this method, the tails of the two vectors Description: 42303.png and Description: 42308.png are placed together, and the diagonal of a parallelogram formed with Description: 42326.png and Description: 42320.png as its sides gives the resultant vector (Figure 1.5).



(i) The magnitude of Description: 36261.png in parallelogram method is given by Description: 36269.png, where θ is the angle between the vectors Description: 42381.png and Description: 39961.png.

(ii) The direction of Description: 42372.png is given by tan Description: 36299.png, where α is the angle that Description: 42390.png makes with Description: 42386.png.



(i) Unit vector: A vector whose magnitude is unity is called unit vector. If Description: 43448.png is a unit vector, then Description: 43444.png = 1.

(ii) Zero vector: A vector whose magnitude is zero is called zero vector (null vector) Description: 43440.png. Thus, Description: 43435.png is a zero vector, if Description: 43430.png = 0.



Negative of a Vector The negative of a vector Description: 42436.png is defined as the vector which, when added to the original vector Description: 42459.png, gives zero for the vector sum, i.e. Description: 42442.png + (–Description: 42448.png) = Description: 40077.png.
Subtraction of Vectors We can subtract vector Description: 42463.png from another vector Description: 42480.png using the definition of negative of a vector as follows:
Description: 42485.png – Description: 42467.png = Description: 42489.png + (– Description: 42472.png).

The geometric construction for the operation of subtracting two vectors is as shown below (Figure 1.6):



Resolution of Vectors into Components

We can also add vectors by taking the projection of a vector along the axes of a rectangular co-ordinate system. These projections are called the components of the vector. Any vector can be completely described by its components.


Let Description: 42508.png be a vector lying in the xy plane making an angle of θ with the positive x-axis as shown in Figure 1.7. The vector Description: 42513.png can be expressed as the sum of two vectors Description: 36472.png and Description: 40258.png is Description: 36480.png.
The component Description: 42563.png represents the projection of Description: 42575.png along the x-axis, and Description: 42568.png represents the projection of Description: 42581.png along the y-axis. From the figure we can see that a right-angled triangle is formed with the three vectors Description: 42587.pngDescription: 42592.png and Description: 42600.png and using the definition of sine and cosine.
we have, Description: 36520.png
Ax = A cos θ
Description: 36531.png
Ay = A sin θ

The magnitude of Description: 42621.png is given by

Description: 36564.png 

and Description: 36574.png

Multiplication of a Vector by a Scalar

When a vector Description: 42645.png is multiplied by a positive scalar quantity n, the product nDescription: 42650.png is a vector having the same direction as Description: 42654.png and magnitude nDescription: 36605.png.

If n is a negative scalar quantity, then the vector nDescription: 42671.png is directed opposite to Description: 42676.png.

Dot Product
Two vectors can be multiplied to obtain either a scalar or a vector. When the product obtained is a scalar, such a multiplication is called a scalar product or a dot product.


Consider two vectors Description: 42690.png and Description: 40472.png as shown in Figure 1.8. Let θ be the angle between them. Then the dot product of Description: 42704.png with Description: 42711.png is defined as
  or Description: 36672.png



The radius of the earth is approximately 6.37 × 106 m. Find

  1. circumference in kilometres,
  2. surface area in square kilometres and
  3. volume in cubic kilometres.

We may assume the shape of the earth to be spherical. Hence, we can apply the formulae related to a sphere. Let R be the radius of the earth.

  1. Circumference of the earth (C) = 2πr
    C = 2π (6.37 × 106 m)
    Description: 37178.png
    Description: 37191.png
    = 4.0 × 104 km
    Surface area of the earth (S) = 4πR2
    S = 4π (6.37 × 106 m)2
    = 5.09 × 1014 m2f
    Description: 37202.png
    Description: 37218.png
    = 5.09 × 108 km2
  2. Volume of the earth (V) = Description: 37231.png
    Description: 37265.png
    Description: 37274.png
    Description: 37281.png
    Description: 37290.png
    = 1.08 × 1012 km3


Estimate the number of heartbeats during a life span of 70 years (assume that normal rate of heartbeat is 70 per min).


Number of heartbeats in one year


Description: 37298.png


Description: 37306.png


= 70 × 60 × 24 × 365 heartbeats/year


Number of heartbeats in 70 years = 70 × 70 × 60 × 24 × 365


= 2.57 × 109 heartbeats


A car is moving with a speed of 40 km h−1 in the northern direction. A bullet is fired from that car with a speed of 30 km h−1 in the western direction. Find the resultant speed of the bullet.


In right-angled triangle OAB,


Description: 37366.png


Description: 37376.png


Description: 37385.png


Description: 37394.png


Description: 37402.png


Resultant speed of the bullet = 50 km h–1



The rectangular components of a force are 3 N and 4 N. What is the magnitude of the force?


P = 3 N, Q = 4 N, θ = 90°


Resultant, Description: 37413.png


Description: 37420.png


Description: 37427.png

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