# Vectors

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 sâˆ’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.
â€‹ 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  is denoted by  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.

â€‹

# 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  and  representing the same physical quantity.

To add a vector  to vector , first draw vector , with its magnitude represented by a convenient scale, and then draw vector  to the same scale with its tail starting from the tip of  as shown in Figure 1.4. The resultant vector  is the vector drawn from the tail of  to the tip of .

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

Note:
(i) The magnitude of  in parallelogram method is given by , where Î¸ is the angle between the vectors  and .

(ii) The direction of  is given by tan , where Î± is the angle that  makes with .

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

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

Negative of a Vector The negative of a vector  is defined as the vector which, when added to the original vector , gives zero for the vector sum, i.e.  + (â€“) = .

Subtraction of Vectors We can subtract vector  from another vector  using the definition of negative of a vector as follows:
â€“  =  + (â€“ ).

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  be a vector lying in the xy plane making an angle of Î¸ with the positive x-axis as shown in Figure 1.7. The vector  can be expressed as the sum of two vectors  and  is .

The component  represents the projection of  along the x-axis, and  represents the projection of  along the y-axis. From the figure we can see that a right-angled triangle is formed with the three vectors  and  and using the definition of sine and cosine.
we have,
âˆ´ Ax = A cos Î¸
Ay = A sin Î¸

The magnitude of  is given by

â€‹

and

# Multiplication of a Vector by a Scalar

When a vector  is multiplied by a positive scalar quantity n, the product n is a vector having the same direction as  and magnitude n.

If n is a negative scalar quantity, then the vector n is directed opposite to .

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  and  as shown in Figure 1.8. Let Î¸ be the angle between them. Then the dot product of  with  is defined as

or

# Illustrations

Example-1

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

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

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) = 2Ï€r

C = 2Ï€ (6.37 Ã— 106 m)

= 4.0 Ã— 104 km

Surface area of the earth (S) = 4Ï€R2

S = 4Ï€ (6.37 Ã— 106 m)2

= 5.09 Ã— 1014 m2f

= 5.09 Ã— 108 km2
2. Volume of the earth (V) =

= 1.08 Ã— 1012 km3

Example-2

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

Solution

Number of heartbeats in one year

= 70 Ã— 60 Ã— 24 Ã— 365 heartbeats/year

Number of heartbeats in 70 years = 70 Ã— 70 Ã— 60 Ã— 24 Ã— 365

= 2.57 Ã— 109 heartbeats

Example-3

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

Solution

In right-angled triangle OAB,

Resultant speed of the bullet = 50 km hâ€“1

Example-4

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

Solution

P = 3 N, Q = 4 N, Î¸ = 90Â°

Resultant,