### What is scalar and vector quantity?

The physical quantity which have only magnitude but no direction, are called scalar quantity.
Mass, length, time, speed, volume, density, pressure, temperature, work, energy, power, electric current, electric charge, electric potential, electric flux etc are the examples of scalar quantity.

### Scalar or Dot Product

Dot product of two vectors A and B is represented by,
A .B = AB cosθ

Where θ is angle between two vectors A and B.

•   If two vectors A and B are parallel, then θ = 0

A.B = AB
For unit vectors,

î . î = ĵ . ĵ = k.k = 1

•   If two vectors A and B are mutually perpendicular, then θ = 90

A.B = 0

For unit vectors,

î . ĵ = ĵ . k = k.i = 0

•   If two vectors A and B are anti parallel, then θ = 0

A.B = - AB

•   Properties of dot product

1.Dot product of two vectors is commutative.

A . B = B . A

2.Dot product is distributive.

A . ( B + C ) = A . B + A . C

•   Dot product of two vectors A and B in component form
A . B = AxBx + AyBy + AzBz

### Cross Product of two vector

Cross product of two $$\vec{A}$$ and $$\vec{B}$$ is represented by,

$$\vec{A} \times \vec{B} = A B \sin \theta \hat{n}$$

Where $$\hat{n}$$ is the unit vector along the resultant vector.

#### If two vectors $$\vec{A}$$ and $$\vec{B}$$ are parallel,

Then $$\theta = 0^o or 180^o$$
So $$\vec{A} \times \vec{B} = 0$$

For unit vectors

$$\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0$$

#### If two vectors $$\vec{A}$$ and $$\vec{B}$$ are perpendicular,

Then $$\theta = 90^0$$
So $$\vec{A} \times \vec{B} = AB \hat{n}$$

For unit vectors
$$\hat{i} \times \hat{j} = \hat{k}$$ , $$\hat{j} \times \hat{k} = \hat{i}$$ , $$\hat{k} \times \hat{i} = \hat{j}$$

#### Properties of cross product

1. Cross product of two vectors in not commutative.
$$\vec{A} \times \vec{B} = - \vec{B} \times \vec{A}$$

2. Cross product is distributive.
$$\vec{A} \times ( \vec{B} + \vec{C} ) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}$$

#### Cross product of two vectors $$\vec{A}$$ and $$\vec{B}$$ in component form

$$\vec{A} \times \vec{B} = ( A_y B_z - A_z B_y ) \hat{i} + ( A_z B_x - A_x B_z ) \hat{j} + ( A_x B_y - A_y B_x ) \hat{k}$$

### How many fundamental and supplementary units in SI?

In SI, there are seven base ( fundamental ) units and two supplementary units.
Base unit

Physical quantity

Unit

Symbol

Mass

Kilogram

kg

Length

Metre

m

Time

Second

s

Temperature

Kelvin

K

Electric current

Ampere

A

Luminous intensity

Candela

cd

Quantity of matter

Mole

mol

Supplementary units

Physical quantity

Unit

Symbol

Plane angle

Solid angle

sr

Conversion factor
Conversion of length

1 centimetre = 10-2 metre
1 millimetre = 10-3metre
1 micrometre = 10-6metre
1 nanometre = 10-9 metre
1 angstrom= 10-10 metre
1 fermi = 10-15 metre
1 kilometre = 103 metre
1 austronomical unit = 1AU=1.496 × 1011 metre
1 light year = 1 ly = 9.461 ×1015metre
1 mile = 1.609 ×103 metre
1 yard = 0.9144 metre
1 inch = 0.0254 metre

Conversion of time

1 mili second= 10-3 second
1 micro second = 10-6 second
1 neno second = 10-9 second
1 hour = 60 minute = 3600 second
1 day = 24 hours =86400 second
1 year = 365 day = 3.156× 107 second
1 sec = 10-8second

Conversion of mass

1 gram = 10 -3 kg
1 quintal = 100 kg
1 tonne = 1000 kg
1 slug = 14.59…

### What is the Universal law of gravitation?

Force of attraction between two masses $$m_1$$ and $$m_2$$ is given by, $$F = \frac{m_1 m_2}{r^2}$$ Where, r is the distance between two masses $$m_1$$ and $$m_2$$. G is a constant, called the Universal gravitational constant. That is called Universal law of gravitation. Acceleration due to gravity ( variation formula ) If a body is falling freely, under the effect of gravity, then the acceleration in the body is called acceleration due to gravity. Variation in acceleration due to gravity with height Acceleration due to gravity at a height h above the surface of earth is, $$g' = g (1 - \frac{2h}{R_e})$$ Where, Re is the radius of the earth. g is the acceleratin due to gravity. Variation in acceleration due to gravity with depth Acceleration due to gravity at a depth d below the surface of earth is, $$g' = g (1 - \frac{d}{R_e})$$ Where, Re is the radius of the earth. g is the acceleratin due to gravity. Variation in acceleration due to gravity with rotatio…

### Differentiation formula

Fomulae of Differentiation d⁄dx (c) = 0   Where c is constant.d⁄dx (cx) = c   Where c is constant.du⁄dt = du⁄dx ⋅ dx⁄dtd⁄dx (u+v) = du⁄dx + dv⁄dxd⁄dx (uv) = u dv⁄dx + v du⁄dxd⁄dx ( xn ) = n xn-1  Where n is real number.d⁄dx un = n un-1du⁄dx   Where u is function of x.d⁄dx sin x = cos x d⁄dx cos x = -sin x d⁄dx tan x = sec2 x d⁄dx cot x = - cosec2 x d⁄dx sec x = tan x sec x d⁄dx cosec x = - cot x cosec x d⁄dx loge x = 1⁄xd⁄dx loge u = 1⁄udu⁄dxd⁄dx ( ex ) = exd⁄dx ( eax ) = a eaxd⁄dx sin ax = a cos ax d⁄dx sin (ax+b) = a cos (ax+b) d⁄dx cos ax = - a sin ax d⁄dx cos (ax+b) = - a sin (ax+b)