# 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 cm

^{3}, 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

^{âˆ’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

- draw a line whose length is proportional to the magnitude of the vector by choosing a suitable scale and
- 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.

* *

# 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.

It is very convenient to add two vectors using a graphical procedure. There are two methods of vector addition:

- triangle method of vector addition and
- parallelogram method of vector addition.

*Triangle Method of Vector Addition***Consider two vectors and representing the same physical quantity.**

*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:**

* *

* *

# 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.

*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 .

*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.

*A*=

_{x}*A*cos

*Î¸*

*A*=

_{y}*A*sin

*Î¸*

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.**

*Î¸*

*be the angle between them. Then the dot product of with is defined as*

# Illustrations

The radius of the earth is approximately 6.37 Ã— 10^{6} m. Find

- circumference in kilometres,
- surface area in square kilometres and
- 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.

- Circumference of the earth (
*C*) = 2*Ï€r**Ï€*(6.37 Ã— 10^{6}m)*=*4.0 Ã— 10^{4}km*S*) = 4Ï€*R*^{2}^{S = 4Ï€ (6.37 Ã— 106 m)2}^{}= 5.09 Ã— 10^{14}m^{2}f^{}^{}^{8}km^{2} ^{}Volume of the earth (V) =^{ }^{12}km^{3}

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

= 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— 10^{9} heartbeats

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

In right-angled triangle *OAB*,

Resultant speed of the bullet = 50 km h^{â€“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, *Î¸* = 90Â°

Resultant,