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UNIT & DIMENSION
VECTORS
AND
BASIC MATHEMATICS

I N D E X
Topic Page No.
UNITS AND DIMENSIONS
1. Physical Quantities 1
2. Set of fundamental quantities 1
3. Derived physical quantities 2
4. Dimensions and dimensional formula 2
5. Principle of homogeneity 3
6. Uses of dimensional analysis 4
7. Limitations of dimensional analysis 8
9. Order of magnitude calculation 9
10. Solved examples 10
VE CTORS
11. Introductrion 13
12. Representation of vectors 13
13. Terminology of vectors 14
14. Angle between vectors 15
15. Lows of addition and subtraction of vectors 16
16. Unit vector 22
17. Co-ordinate system 23
18. Concept of equilibrium 27
19. Position vector 28
UNIT & DIMENSION
VECTORS
AND
BASIC MATHEMATICS

20. Displacement vector 29
21. Product of vectors 31
22. Solved Examples 38
BASIC MATHEMATICS
23. Trigonometry 44
24. Calculus 48
25. Solved Examples 60
UNITS AND DIMENSIONS
26. Exercises 62
27. Answer Key 68
VE CTORS
28. Exercises 69
29. Answer Key 77
BASIC MATHEMATICS
30. Exercises 78
31. Answer Key 79
32. Hints & Solutions 80

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AC C- PH-U NIT DIMENSIO N 1
UNITS AND DIMENSIONS
PHYSICAL QUANTITIES
All quantities that can be measured are called physical quantities. eg. time, length, mass, force, work
done, etc. In physics we study about physical quantities and their inter relationships.
MEASUREMENT
Measurement is the comparison of a quantitywith a standard of the same physical quantity.
UNITS
All physical quantities are measured w.r.t. standard magnitude of the same physical quantity and these
standards are called UNITS. eg. second, meter, kilogram, etc.
So the four basic properties of units are:—
1. They must be well defined.
2. They should be easily available and reproducible.
3. They should be invariable e.g. step as a unit of length is not invariable.
4. They should be accepted to all.
SET OF FUNDAMENTAL QUANTITIES
A set of physical quanties which are completelyindependent of each other and all other physical quantities
can be expressed in terms of these physical quantities is called Set of Fundamental Quantities.
Physical Quantity Units(SI) Units(CGS) Notations
Mass kg (kilogram) g M
Length m (meter) cm L
Time s (second) s T
Temperature K (kelvin) °C 
Current A (ampere) A I or A
Luminous intensity cd (candela) — cd
Amount of substance mol — mol
Physical Quantity Definition
(SI Unit)
Length (m) The distance travelled by light in vacuum in 458,792,299
1
second is called 1 metre.
Mass (kg) The mass of a cylinder made of platinum-iridium alloykept
at International Bureau of Weights and Measures is de-
fined as 1 kilogram.
Time (s) The second is the duration of 9,192,631,770 periods of

2AC C- PH-U NIT DIMENSIO N
BANSAL CLASSES Private Ltd. ‘Gaurav Tower’, A-10, Road No.-1, I.P.I.A., Kota-05
the radiation corresponding to the transition between the
two hyperfine levels of the ground state of the cesium-
133 atom.
Electric Current (A) If equal currents are maintained in the two parallel infinitelylong
wires ofnegligible cross-section, so that the force between
them is 2 × 10–7 newton per metre of the wires, the current in
any of the wires is called 1Ampere.
Thermodynamic Temperature (K) The fraction
16
.
273
1of the thermodynamic temperature
of triple point of water is called 1 Kelvin
Luminous Intensity(cd) 1 candela is the luminous intensity of a blackbody of
surface area 2
m
000,600
1placed at the temperature of
freezing platinum and at a pressure of 101,325 N/m2, in
the direction perpendicular to its surface.
Amount of substance (mole) The mole is the amount of a substance that contains as
many elementary entities as there are number of atoms in
0.012 kg of carbon-12.
There are two supplementary units too:
1. Plane angle (radian) angle = arc / radius
r
=/ r
2. SolidAngle (steradian)
DERIVED PHYSICAL QUANTITIES
The physical quantities those can be expressed in terms of fundamental physical quantities are called
derived physical quantities.eg. speed = distance/time.
DIMENSIONS AND DIMENSIONAL FORMULA
All the physical quantities of interest can be derived from the base quantities.
DIMENSION
The power (exponent) of base quantitythat enters into the expression of a physical quantity, is called the
dimension of the quantity in that base.
To make it clear, consider the physical quantity"force".
Force = mass × acceleration
=
time
time/length
mass 
= mass × length × (time)–2
So the dimensions of force are 1 in mass, 1 in length and –2 in time. Thus
[Force] = MLT–2