Chapter 1 — Units, Dimensions and Vectors

Class 12th NIOS Physics 312 Chapter 1 notes

1. Physical world and measurements

Scope of Physics

Physics studies natural phenomena across many scales — from subatomic particles to the universe. It includes mechanics, thermodynamics, waves and optics, electromagnetism, atomic and nuclear physics, and applied branches such as biophysics and astrophysics. Physics underlies modern technology and engineering.

Nature of physical laws

• Formed from repeated experiments and observations.
• Typically universal, simple, stable (within their domain) and mathematically expressible.
• Used to predict and describe natural phenomena.

Physics, technology and society

Applications of physics have produced engines (thermodynamics), communication systems (electromagnetic waves), electrical generation (electromagnetic induction), nuclear reactors, aircraft and rockets (Newton’s laws), medical imaging (X-rays, lasers), and electronics (semiconductors).

Need for measurement

Measurements are essential to quantify observations and to allow reproducibility. Standard units and agreed conventions are necessary for clear communication of results.

2. Unit of measurement

SI base units

The International System of Units (SI) defines seven base units:

Prefixes for powers of ten

Multiples and submultiples of SI units use prefixes. Common prefixes:

Standards of mass, length and time

• Mass: kilogram defined by an international prototype (historically a platinum-iridium cylinder); national prototypes are maintained by national laboratories.
• Length: metre defined as the distance light travels in vacuum in 1/299,792,458 second (i.e., c = 299,792,458 m/s is fixed).
• Time: second defined by the radiation frequency of the cesium-133 atom: 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of its ground state.

Role of precise measurements

Precision can reveal new phenomena and lead to discoveries (e.g., Rayleigh’s argon discovery; Michelson–Morley experiment that contributed to relativity). Modern atomic clocks reach uncertainties of parts in 10^-15 or better.

3. Significant figures

Definition

Significant figures are the digits in a measurement that are known with certainty plus the first uncertain digit. They reflect the precision of the measurement.

Rules for counting significant figures

1. All non-zero digits are significant. (e.g., 315.58 → 5 s.f.)
2. Zeros between non-zero digits are significant. (e.g., 5300405.003 → 10 s.f.)
3. Zeros to the right of a decimal point and to the right of a non-zero digit are significant. (e.g., 50.00 → 4 s.f.)
4. Leading zeros in a decimal fraction are not significant. (e.g., 0.00043 → 2 s.f.)
5. Trailing zeros in an integer that come from measurement are significant. (e.g., 4050 m measured to nearest metre → 4 s.f.)
6. In a pure counting or defined numbers (like 100 people), the zeros are not significant unless specified.

Operations with significant figures

• Addition/Subtraction: limit the result to the least precise decimal place among the operands.
• Multiplication/Division: limit the result to the least number of significant figures among the operands.

4. Derived units

Derived units are combinations of base units. Examples with special names:

Nomenclature and symbols (good practice)

• Unit symbols remain the same in plural (e.g., 5 m, not 5 ms or 5 ms.).
• Write symbols without full stop: cm not cm.
• Avoid double prefixes (use ns for nanosecond, not mμs).
• When combining prefix and unit, treat as a single symbol: μs^-1 not (10^-6s)^-1.
• When writing unit names in sentence form, use lowercase (except Kelvin, which is ‘kelvin’ but symbol ‘K’).

5. Dimensions of physical quantities

Dimensional notation

Dimensions express how a quantity depends on base quantities. Common base dimensions (mechanics) are:

• Mass: M
• Length: L
• Time: T

Examples:

• Volume = L³
• Density = M·L^-3
• Speed = L·T^-1
• Acceleration = L·T^-2
• Force = M·L·T^-2

Dimensional analysis and principle of homogeneity

All terms in a physical equation must have the same dimensions. Dimensional analysis is used to:

1. Check the correctness of equations.
2. Derive relations (up to a dimensionless constant) between quantities.
3. Convert between unit systems.
4. Obtain dimensions of unknown quantities.

6. Vectors and scalars

Scalars

Quantities described by magnitude only: examples include mass, temperature, energy, speed (magnitude only).

Vectors

Quantities that require magnitude and direction: examples include displacement, velocity, acceleration, force, momentum.

Representation of vectors

A vector is represented by an arrow. Notation:

• Vector A often written as <vector>A or bold A; magnitude denoted |A| or A.
• Unit vector in direction of A: n̂ (hat indicates unit vector).

Addition of vectors

Two graphical methods:

1. Triangle law: Place tail of B at head of A. Resultant R is from tail of A to head of B.
2. Parallelogram law: Complete the parallelogram with sides A and B; diagonal is resultant R.

Parallelogram formula (magnitude)

For vectors A and B with angle θ between them:

Direction (angle α made by resultant with A):

Special cases

• If θ = 0 (parallel) → R = A + B (same direction).
• If θ = 180° (anti-parallel) → R = |A – B| (direction of larger vector).
• If θ = 90° (perpendicular) → R = √(A² + B²).

Subtraction of vectors

A − B = A + (−B). Graphically, reverse B and then add.

7. Multiplication of vectors

Scalar multiplication

Multiplying a vector A by scalar k gives vector kA with magnitude |k|·|A|. If k negative, direction reverses.

Dot product (scalar product)

Definition: A · B = AB cos θ (θ is angle between A and B). Result is a scalar.

• Properties: commutative (A·B = B·A), distributive.
• Work done by force F through displacement d: W = F·d = F d cos θ.

Cross product (vector product)

Definition: A × B is a vector of magnitude AB sin θ and direction perpendicular to the plane of A and B (use right-hand rule). It is anti-commutative: A × B = −(B × A).

Unit vectors and components

Unit vectors along axes: î (x-axis), ĵ (y-axis), k̂ (z-axis). A vector A with components A_x, A_y, A_z:

Dot product in components:

Cross product in component (determinant) form:

A × B = |  î     ĵ     k̂  |
        | A_x  A_y  A_z |
        | B_x  B_y  B_z |

Cross products of unit vectors

8. Resolution of vectors

To get components of a vector A that makes angle θ with x-axis:

If components A_x and A_y are known:

Worked examples

Dimensions: L = (L T^-2)^n (T)^m = L^n T^(-2n+m)
Equate powers: for L: 1 = n ; for T: 0 = -2n + m  ⇒ m = 2
Thus x ∝ a^1 t^2 ⇒ x ∝ a t^2
Complete relation (from kinematics): x = (1/2) a t^2

R = √(70^2 + 50^2 + 2·70·50 cos 135°)
cos 135° = -cos 45° = -√2/2
Compute: R ≈ 49.5 N
Direction α from A: tan α = (B sinθ) / (A + B cosθ)
After substitution α ≈ 45° (north-west)

9. Summary — key points

• Always state a measured quantity with units; SI units are the accepted standard for science.
• Significant figures indicate measurement precision; follow rules when calculating results.
• Dimensions provide a way to check equations and derive relationships (principle of homogeneity).
• Distinguish scalar and vector quantities; handle vector addition, subtraction, scalar and vector products properly.
• Resolve vectors into components to simplify problems in mechanics and electromagnetism.

10. In-text questions and terminal exercises (practice)

11. Answers / Hints to selected problems

Easy Learning Tricks for Chapter 1

1. SI Base Units — “L M T A K C Mo” Trick

To remember all seven SI base quantities:

“Love Makes Time Always Keep Coming More.”
→ L – Length (metre)
→ M – Mass (kilogram)
→ T – Time (second)
→ A – Current (ampere)
→ K – Temperature (kelvin)
→ C – Luminous Intensity (candela)
→ Mo – Amount of substance (mole)

2. Dimensional Formula Quick Recall — “MaLiTa” Method

Write everything as a combination of M, L, and T.
Some common ones to memorize:

Mnemonic for sequence:
“Force Presses Energy Powerfully” → Force → Pressure → Energy → Power

3. Significant Figures Rules – “NIZT”

Remember this short code:
N I Z T → Non-zero, Internal zeros, Zeros after decimal, Trailing zeros (only if measured)

Example logic:

• N: Non-zero digits always count

• I: Zeros inside count (405 → 3 s.f.)

• Z: Zeros after decimal count (45.00 → 4 s.f.)

• T: Trailing zeros in integers count only if specified (5000 → ambiguous)

4. Dimensional Analysis — Uses Memory Trick “V-C-U-F”

Applications of dimensional equations:

1. V – Verify correctness of formulas

2. C – Convert units

3. U – Derive relations (unknown powers)

4. F – Find dimensions of constants

Remember: “Dimensional Analysis is Very Clever Useful Formula-checker.”

5. Vector Laws Mnemonic — “TPR”

For vector addition and subtraction, recall:

T – Triangle Law
P – Parallelogram Law
R – Resultant

Formula recall trick:
“Cos joins, Sin separates.”
→ Use cos θ when vectors are combined (R = √(A² + B² + 2AB cos θ)).
→ Use sin θ for perpendicular components or vector products.

6. Cross Product Orientation — “Right-hand Rule”

Curl your right-hand fingers from first vector → second vector,
thumb points in direction of resultant (perpendicular vector).
Remember: Cross → Curl → Thumb.

7. Unit Vector Components — “i j k XYZ”

For any 3D vector:
A = Aₓ î + Aᵧ ĵ + A𝓏 k̂

Mnemonic: “I Jump Kross” (i–j–k order)
and in determinant form, always place i, j, k on top.

8. Dimensional Homogeneity Check Shortcut

Whenever you see √, sin, cos, log, or e, the argument inside must be dimensionless.
Example: sin θ (θ = dimensionless); √(a/g) (both have same dimension).

9. Common Errors to Avoid

• Writing “sec” instead of “s” for second (confused with trigonometric sec).

• Mixing prefixes (write 1 ns, not 0.001 μs).

• Forgetting that SI unit symbols are never plural or dotted (e.g., m not m.).

10. Quick Formula Snapshot Table

11. Short Story to Remember Vectors

Think of Force and Direction as Friends —
they always go together (vectors).
But Mass and Energy are loners —
they only have magnitude (scalars).