Unit & Dimension, Basic Maths & Vectors

Physics is built on measurement. Before we can write equations, describe motion, or calculate energy, we need a shared language of units. The International System of Units (SI) is that language, and it defines seven independent base quantities from which every other physical quantity can be derived.

The seven SI base quantities and their units are:

Every other physical quantity — force, energy, pressure, power — is derived from these seven. A derived unit is simply a combination of base units raised to various powers. For example, speed = distance/time gives the unit m/s, which is L/T = $[LT^{-1}]$.

The dimensional formula of a physical quantity expresses it in terms of the base quantities M (mass), L (length), and T (time), with each raised to some power. Writing these formulas is not about memorising a table — it is about understanding the definition of each quantity and following it through.

Here is a systematic derivation of the most important quantities:

Notice that momentum and impulse share the same dimensional formula. This is physically meaningful — impulse equals change in momentum (Newton's second law in integral form).

Notice: torque and work have the same dimensional formula $[ML^2T^{-2}]$, yet they are completely different physical quantities. This is a common JEE trap — two quantities sharing dimensions does not make them identical.

Dimensional analysis is a powerful tool that uses the fact that every valid physical equation must be dimensionally homogeneous — every term must have the same dimensional formula. This gives us three major applications.

Application 1 — Checking consistency of an equation

Consider the equation $v^2 = u^2 + 2as$. Left side: $[v^2] = [L^2T^{-2}]$. Right side: $[u^2] = [L^2T^{-2}]$ and $[2as] = [LT^{-2}][L] = [L^2T^{-2}]$. All terms match, so the equation is dimensionally consistent.

Application 2 — Converting between unit systems

The rule is: numerical value × unit = constant (the physical reality doesn't change). If a quantity has dimensions $[M^aL^bT^c]$, the conversion factor is:

Example: Convert 1 joule to CGS. Energy has dimensions $[ML^2T^{-2}]$. Using $M_1/M_2 = 1000$ g/g, $L_1/L_2 = 100$ cm/cm, $T_1/T_2 = 1$: $n_2 = 1 \times 1000 \times 100^2 = 10^7$ ergs.

Application 3 — Deriving relationships between quantities

Suppose the time period T of a simple pendulum depends on its length L, mass m, and gravitational acceleration g. Assume $T = k\, L^a m^b g^c$. Matching dimensions on both sides of the equation:

So $T \propto \sqrt{L/g}$. Dimensional analysis gives the correct structure, but the constant $k = 2\pi$ must come from experiment or full derivation.

A working knowledge of trigonometry is indispensable for JEE Physics. Every topic from inclined planes to projectile motion to optics relies on decomposing vectors and resolving forces using trigonometric ratios.

Standard Angle Table

The pair 37°–53° appears constantly in JEE because it comes from the 3-4-5 right triangle. If the sides opposite, adjacent, and hypotenuse are 3, 4, 5, then the angle at the base is 37° (sin 37° = 3/5, cos 37° = 4/5) and the complementary angle is 53°.

Small Angle Approximation

When θ is very small (much less than 1 radian, roughly less than 10°), the first-order Taylor expansion gives:

This approximation is used throughout physics — in the derivation of simple pendulum equations, in optics for paraxial rays, in waves for small oscillations.

Binomial Approximation

When $|x| \ll 1$, the binomial theorem gives:

This is extremely useful when a physical quantity changes by a small fraction. For example, $g' = g(1 + h/R)^{-2} \approx g(1 - 2h/R)$ when $h \ll R$. The key skill is recognising when to rewrite an expression in $(1+x)^n$ form.

Physical quantities split into two categories based on whether they require a direction to be fully described. Scalars — mass, temperature, energy, speed — are fully described by magnitude alone. Vectors — displacement, velocity, force, acceleration, momentum — require both magnitude and direction.

A vector is conventionally written with an arrow overhead ($\vec{A}$) or in bold (**A**). Its magnitude is written as $|\vec{A}|$ or simply A. A vector multiplied by a positive scalar changes magnitude; multiplied by a negative scalar, it also reverses direction.

Unit Vector

The unit vector in the direction of $\vec{A}$ is obtained by dividing $\vec{A}$ by its own magnitude:

The standard Cartesian unit vectors are $\hat{i}$ (along +x), $\hat{j}$ (along +y), and $\hat{k}$ (along +z). They are mutually perpendicular and each has magnitude 1.

Triangle Law of Vector Addition

To add $\vec{A}$ and $\vec{B}$: place the tail of $\vec{B}$ at the tip of $\vec{A}$. The resultant $\vec{R} = \vec{A} + \vec{B}$ is the vector from the tail of $\vec{A}$ to the tip of $\vec{B}$.

Parallelogram Law

Place both vectors at the same origin. The diagonal of the parallelogram they form is the resultant. If $\theta$ is the angle between $\vec{A}$ and $\vec{B}$:

The direction of the resultant with respect to $\vec{A}$ is given by:

When two vectors are multiplied, the result can be either a scalar (dot product) or a vector (cross product), depending on the physical context.

Scalar (Dot) Product

The dot product is the product of the magnitudes and the cosine of the angle between them. It equals zero when the vectors are perpendicular ($\theta = 90°$). It is maximum (AB) when they are parallel ($\theta = 0°$) and minimum (−AB) when antiparallel ($\theta = 180°$).

In Cartesian form: $\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z$.

Physical examples: Work = $\vec{F} \cdot \vec{d}$, Power = $\vec{F} \cdot \vec{v}$.

Vector (Cross) Product

The direction of $\vec{A} \times \vec{B}$ is perpendicular to the plane containing $\vec{A}$ and $\vec{B}$, given by the right-hand rule. The cross product is zero when vectors are parallel or antiparallel.

In Cartesian form using the determinant:

Physical examples: Torque = $\vec{r} \times \vec{F}$, Angular momentum = $\vec{r} \times \vec{p}$.

Cartesian Resolution

Any vector $\vec{A}$ can be written as a sum of its Cartesian components:

For a 2D vector at angle θ with the x-axis: $A_x = A\cos\theta$, $A_y = A\sin\theta$.