Basic Mathematics & Vectors

Trigonometry is the backbone of NEET Physics. Vector resolution, projectile motion, circular motion, waves, and optics all depend on your ability to read and apply sine, cosine, and tangent ratios instantly.

Standard Angle Values

Memory trick for sin: sin 0°, 30°, 45°, 60°, 90° = $\sqrt{0}/2,\; \sqrt{1}/2,\; \sqrt{2}/2,\; \sqrt{3}/2,\; \sqrt{4}/2$. For cos, read the table backwards.

Key Identities

Small Angle Approximation

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

NEET Physics requires calculus at the conceptual level — you need to recognise when a quantity is a derivative or an integral of another, and apply a small set of standard rules. You do not need to solve complex integrals; you need to understand what derivatives and integrals mean physically.

Standard Derivatives

Standard Integrals

Physical Meaning of Derivatives

• Velocity $= \dfrac{ds}{dt}$ — rate of change of displacement with time
• Acceleration $= \dfrac{dv}{dt} = \dfrac{d^2s}{dt^2}$ — rate of change of velocity
• The slope of a displacement–time graph gives instantaneous velocity.
• The slope of a velocity–time graph gives instantaneous acceleration.

Physical Meaning of Integration

• $s = \int v\,dt$ — displacement is the area under the $v$–$t$ graph.
• $v = \int a\,dt$ — change in velocity is the area under the $a$–$t$ graph.
• Work done = $\int \vec{F} \cdot d\vec{s}$ — area under the $F$–$s$ graph.

Quadratic Formula

For the equation $ax^2 + bx + c = 0$, the roots are:

The discriminant $D = b^2 - 4ac$ tells you the nature of roots:

• $D > 0$ — two distinct real roots (two physically possible answers; discard unphysical ones, e.g., negative time)
• $D = 0$ — one repeated real root
• $D < 0$ — no real roots (the physical scenario is impossible with the given values)

Binomial Approximation

When $|x| \ll 1$ (x is much smaller than 1), the binomial theorem simplifies to:

Common forms you should recognise instantly:

Physics example — gravity at height h:

Every physical quantity is either a scalar or a vector. Getting this distinction right prevents a large class of conceptual errors in NEET.

Scalars — fully described by magnitude alone. No direction needed.

Examples: mass, temperature, energy, work, power, time, speed, electric charge, pressure, density.

Vectors — require both magnitude and direction for complete description.

Examples: displacement, velocity, acceleration, force, momentum, electric field, magnetic field, torque, angular momentum, weight.

Unit Vectors

A unit vector has magnitude exactly 1. It specifies direction only.

The Cartesian unit vectors are $\hat{i}$ (along +x), $\hat{j}$ (along +y), $\hat{k}$ (along +z). They are mutually perpendicular: $|\hat{i}| = |\hat{j}| = |\hat{k}| = 1$.

Special Vectors

• Equal vectors — same magnitude and same direction (position doesn't matter).
• Negative vector of $\vec{A}$ is $-\vec{A}$: same magnitude, opposite direction.
• Zero (null) vector $\vec{0}$: magnitude zero, direction undefined. Example: displacement after returning to starting point.
• Position vector $\vec{r}$: locates a point relative to the origin. In 2D: $\vec{r} = x\hat{i} + y\hat{j}$.

Triangle Law of Vector Addition

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 $\vec{B}$. This works for any number of vectors placed head-to-tail.

Parallelogram Law

Place both $\vec{A}$ and $\vec{B}$ at the same origin. Complete the parallelogram. The diagonal from the origin is $\vec{R}$.

where $\theta$ = angle between $\vec{A}$ and $\vec{B}$; $\varphi$ = angle of $\vec{R}$ with $\vec{A}$.

Special cases of resultant magnitude:

Vector Subtraction

$\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$. Reverse the direction of $\vec{B}$ and then add. The magnitude of $\vec{A} - \vec{B}$ is:

Any vector can be split into components along chosen directions — usually the x and y axes. This is called resolution or decomposition of a vector. It is the most-used skill in all of mechanics.

Cartesian Components

For a vector $\vec{A}$ of magnitude $A$ making angle $\theta$ with the positive x-axis:

Adding vectors by components

To add $\vec{A}$ and $\vec{B}$: add their x-components and y-components separately.

Three-dimensional vectors

The dot product (scalar product) of two vectors $\vec{A}$ and $\vec{B}$ is a scalar quantity defined as:

where $\theta$ is the angle between the two vectors when placed tail-to-tail.

Properties of Dot Product

• Commutative: $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$
• Distributive: $\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A}\cdot\vec{B} + \vec{A}\cdot\vec{C}$
• Self dot product: $\vec{A}\cdot\vec{A} = A^2$

Cartesian unit vector results:

Component form:

Finding angle between two vectors:

Physical Applications

• Work: $W = \vec{F}\cdot\vec{d} = Fd\cos\theta$ — zero when force is perpendicular to displacement
• Power: $P = \vec{F}\cdot\vec{v} = Fv\cos\theta$
• Electric flux: $\Phi_E = \vec{E}\cdot\vec{A}$ (area vector)
• Magnetic flux: $\Phi_B = \vec{B}\cdot\vec{A}$

The cross product (vector product) of two vectors $\vec{A}$ and $\vec{B}$ is a vector quantity:

The direction is perpendicular to the plane of $\vec{A}$ and $\vec{B}$, given by the right-hand rule.

Right-Hand Rule: Point fingers of the right hand along $\vec{A}$, curl them towards $\vec{B}$. The thumb points in the direction of $\vec{A}\times\vec{B}$.

Properties

• Anti-commutative: $\vec{A}\times\vec{B} = -(\vec{B}\times\vec{A})$
• Self cross product: $\vec{A}\times\vec{A} = \vec{0}$ (magnitude = $A^2\sin 0° = 0$)
• Parallel or anti-parallel vectors: $\vec{A}\times\vec{B} = \vec{0}$

Cartesian unit vector results (cyclic rule):

Determinant formula:

Physical Applications

• Torque: $\vec{\tau} = \vec{r}\times\vec{F}$ — magnitude = $rF\sin\theta$, direction by right-hand rule
• Angular momentum: $\vec{L} = \vec{r}\times\vec{p} = \vec{r}\times m\vec{v}$
• Magnetic force: $\vec{F} = q(\vec{v}\times\vec{B})$ — cross product gives the direction of the Lorentz force
• Area of parallelogram formed by $\vec{A}$ and $\vec{B}$: $= |\vec{A}\times\vec{B}|$

These are the most commonly tested vector concepts in NEET, along with the exact mistakes students make and how to avoid them.

Trap 1 — Maximum and minimum resultant

Any resultant between $|A-B|$ and $A+B$ is possible. NEET questions like "can the resultant be 7 if A=3 and B=5?" — answer: yes ($|3-5|=2 \leq 7 \leq 8=3+5$). "Can it be 10?" — no.

Trap 2 — Dot product zero does not mean one vector is zero

$\vec{A}\cdot\vec{B} = 0$ means $\theta = 90°$ (perpendicular), not that either vector is zero. NEET options sometimes include "one of the vectors is a null vector" — this is wrong unless stated.

Trap 3 — Cross product of parallel vectors is zero

If $\vec{A}\times\vec{B} = \vec{0}$ and neither is a zero vector, the vectors are parallel or anti-parallel ($\theta = 0°$ or $180°$). Example: a force parallel to displacement does zero torque.

Trap 4 — Null vector subtleties

• $\vec{A} + (-\vec{A}) = \vec{0}$ — valid. But $\vec{A} - \vec{A} = \vec{0}$ only when both are the same vector.
• A null vector has zero magnitude but its direction is indeterminate — not "in all directions".
• $\vec{A}\times\vec{A} = \vec{0}$ always (angle between a vector and itself is 0°, sin 0° = 0).

Trap 5 — Component confusion

• The component of $\vec{A}$ along $\vec{B}$ is $A\cos\theta$ (a scalar). The vector component is $A\cos\theta\,\hat{B}$.
• A component can be larger than the vector if the angle is imaginary — impossible. Components are always $\leq$ magnitude.
• If $A_x = A$, then $A_y = 0$ ($\theta = 0°$, vector along x-axis).

Use this as a rapid revision reference before the exam. Every formula here has appeared in NEET at least once in the past five years.

Trigonometry

Calculus

Vector Operations