Mathematical Tools — NEET Notes

Scalars and vectors — the distinction that runs through physics

Every physical quantity falls into one of two families. A scalar has only a magnitude — a number with a unit. Mass, length, time, energy, temperature, electric charge, and density are all scalars. You can add them the way you add ordinary numbers. A vector has both a magnitude and a direction in space. Displacement, velocity, acceleration, force, momentum, torque, electric field, and magnetic field are all vectors. You cannot add vectors arithmetically — the direction matters and changes the answer.

A vector is drawn as a directed line segment: its length encodes the magnitude and its arrow encodes the direction. In symbols we write A in bold (or as A with an arrow over it) and |A| or A for its magnitude. A vector with magnitude one is a unit vector, written with a hat: î, ĵ, k̂ are the unit vectors along x, y, and z. Two vectors are equal only when both their magnitude and direction match — a parallel translation does not change a vector, but rotating it does.

Addition and subtraction of vectors

To add two vectors A and B, place the tail of B at the tip of A; the resultant R is the vector from the tail of A to the tip of B. This is the triangle law of vector addition. Equivalently, place both tails at a common point and complete the parallelogram — the diagonal from the common point is the resultant. This is the parallelogram law. Both rules give the same answer; you choose whichever is geometrically more convenient.

If two vectors of magnitudes A and B make an angle θ with each other, the magnitude of their resultant is given by the parallelogram-law formula. Vector addition is commutative (A + B = B + A) and associative (A + (B + C) = (A + B) + C). To subtract, reverse the vector being subtracted and add: A − B = A + (−B). The negative of a vector has the same magnitude and the opposite direction.

|R| = √(A² + B² + 2AB cos θ)

Magnitude of the resultant when A and B make angle θ

The formula collapses neatly at the limits. When θ = 0 (vectors parallel), |R| = A + B — the maximum. When θ = 180° (vectors antiparallel), |R| = |A − B| — the minimum. When θ = 90° (vectors perpendicular), the cross term vanishes and |R| = √(A² + B²) — the Pythagorean case that NEET reaches for in projectile motion, electric field superposition, and force-balance problems.

Resolving a vector into components

Adding vectors graphically works in two dimensions but becomes unwieldy in three, and it cannot survive being typed into an exam answer. The standard workaround is resolution: replace any vector by its perpendicular projections onto the coordinate axes — its components. A vector in the xy-plane that makes angle θ with the +x axis has x-component Aₓ = A cos θ and y-component A_y = A sin θ. The original vector is recovered as A = Aₓ î + A_y ĵ, and its magnitude is A = √(Aₓ² + A_y²) with direction tan θ = A_y/Aₓ.

Once every vector in a problem is resolved into components, addition reduces to ordinary arithmetic on each axis separately — no parallelograms, no triangles. Add the x-components together, add the y-components together, and reassemble. This is the workflow you use for inclined-plane problems, projectile motion, and equilibrium of concurrent forces. It generalises to three dimensions by adding a z-component along k̂.

Dot product and cross product

Vectors can be multiplied in two genuinely different ways. The dot product (also called the scalar product) returns a number; the cross product (also called the vector product) returns a new vector. Both appear everywhere in physics, and choosing the wrong one is the single most expensive maths error in NEET Physics.

The dot product is defined as A · B = AB cos θ, where θ is the angle between the two vectors. It measures how much one vector lies along the other; if they are perpendicular, the dot product is zero. The cross product is defined as A × B = AB sin θ n̂, where n̂ is a unit vector perpendicular to both A and B, chosen by the right-hand rule — curl the fingers of your right hand from A to B; your thumb points along A × B. Cross products are not commutative: A × B = −(B × A).

In component form the calculations are mechanical. If A = Aₓî + A_y ĵ + A_z k̂ and B = Bₓî + B_y ĵ + B_z k̂, then A · B = AₓBₓ + A_y B_y + A_z B_z. The cross product is the 3×3 determinant of the matrix with rows î, ĵ, k̂; (Aₓ, A_y, A_z); (Bₓ, B_y, B_z). The two diagnostic checks every NEET aspirant should run before solving any product problem: is the answer a number or a vector? and does the answer vanish when the vectors line up, or when they cross perpendicularly? One question, and you know which product you need.

Differentiation — the rate-of-change machine

Most physics quantities are rates: velocity is the rate at which position changes, acceleration is the rate at which velocity changes, current is the rate at which charge flows, power is the rate at which energy is transferred. The derivative dy/dx of a function y(x) measures exactly this — how fast y changes as x changes. Geometrically, it is the slope of the tangent line to the curve y = f(x) at a chosen point. The same symbol covers position vs time (giving velocity), velocity vs time (giving acceleration), and charge vs time (giving current).

You do not derive derivatives from first principles in NEET; you memorise a small table and combine them with three rules — the sum rule (d/dx of a sum is the sum of the derivatives), the product rule (d/dx[uv] = u′v + uv′), and the chain rule (d/dx of a function-of-a-function — see the pull quote below). Once the table is in your bones, finding velocity from a position-time equation or current from a charge-time equation takes seconds.

If y depends on u and u depends on x, then dy/dx = (dy/du)(du/dx).

The chain rule — the rule that unlocks every composite function

The chain rule is the workhorse. Almost every derivative you take in physics is a chain-rule derivative, even when it does not look like one. d/dt[sin(ωt)] = ω cos(ωt) — the inside is ωt, so its derivative ω is pulled out. d/dx[(2x + 1)⁵] = 5(2x + 1)⁴ · 2. d/dt[e^(−λt)] = −λe^(−λt) — the decay constant comes out front. Whenever you see anything other than a bare x inside a function, the chain rule is waiting.

Worked examples — physics in derivative form

If a particle's position is x(t) = 4t² − 3t + 7, its velocity is v = dx/dt = 8t − 3 and its acceleration is a = dv/dt = 8 m/s². The 7 vanishes (constant), the −3t becomes −3, the 4t² becomes 8t. If charge on a capacitor varies as q(t) = q₀(1 − e^(−t/τ)), the current is i = dq/dt = (q₀/τ) e^(−t/τ) — the chain rule pulls the −1/τ out, and the minus signs cancel. These are not novel calculations; they are the same table applied with care.

Integration — the inverse machine

If differentiation extracts rates, integration rebuilds totals from rates. Given v(t), the position is x(t) = ∫v(t) dt + C. Given F(x), the work done from x = a to x = b is ∫ₐᵇ F(x) dx. Integration is the inverse of differentiation: if dF/dx = f(x), then ∫f(x) dx = F(x) + C. The arbitrary constant C reminds you that any constant has zero derivative, so the antiderivative is determined only up to that constant — set by an initial condition such as "the particle was at x₀ at t = 0".

There are two flavours. An indefinite integral ∫f(x) dx is a function of x plus a constant — a family of antiderivatives. A definite integral ∫ₐᵇ f(x) dx is a single number, equal to F(b) − F(a) by the fundamental theorem of calculus. Geometrically, the definite integral is the signed area under the curve y = f(x) between x = a and x = b — areas below the axis count negative. This area interpretation is why work, charge, impulse, and displacement all show up as integrals.

Two physics workflows recur constantly. Kinematics from acceleration: if a(t) is given, integrate once to get v(t) (fix C using v(0)), integrate again to get x(t) (fix C using x(0)). Work from a variable force: if F(x) is the force, W = ∫F(x) dx between the start and end positions; the same construction gives the area-under-the-graph interpretation of work. The definite integral always returns a number — no constant C — because the constant cancels between the upper and lower limits.

Trigonometry and small-angle approximations

Trigonometry enters physics through every angle. Resolution of vectors, oscillations of springs and pendulums, wave equations, optics with prisms and lenses — all of them sit on the three core functions sin θ, cos θ, tan θ. The two basic identities you must know cold are the Pythagorean identity sin²θ + cos²θ = 1 and the double-angle identities sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ − sin²θ = 1 − 2 sin²θ = 2 cos²θ − 1. The last two forms are what let you rewrite sin²θ as (1 − cos 2θ)/2 — the trick that integrates sin²x and gives the RMS value of an AC sinusoid.

Angles in physics are always in radians, not degrees, whenever calculus is involved — d/dx(sin x) = cos x only when x is in radians. The conversion is 180° = π radians. Standard values: sin 0 = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1. Cosine reads the same values backwards. NEET expects these to be at fingertip speed.

Why does small-angle matter so much? Because the equation of motion of a pendulum is exactly mℓθ̈ = −mg sin θ — which is not the equation of simple harmonic motion. Replace sin θ by θ (valid for small swings), and it becomes θ̈ = −(g/ℓ)θ, the canonical SHM equation with angular frequency ω = √(g/ℓ). The same approximation turns Snell's law n₁ sin θ₁ = n₂ sin θ₂ into n₁θ₁ = n₂θ₂ for paraxial rays in geometrical optics. Most "exam-friendly" physics formulas are small-angle approximations of more complicated truths.

Logarithms and exponentials

The exponential function eˣ and its inverse, the natural logarithm ln x, run through every decay and growth problem in physics. Radioactive decay follows N(t) = N₀ e^(−λt). Capacitor charging follows q(t) = q₀(1 − e^(−t/RC)). Beer-Lambert absorption is I = I₀ e^(−μx). All three share the same shape and yield to the same trick: take the natural log of both sides to linearise — ln N = ln N₀ − λt — turning an exponential decay into a straight line with slope −λ.

The two log laws every NEET aspirant uses are ln(ab) = ln a + ln b (the log of a product is the sum of the logs) and ln(a^n) = n ln a (a power becomes a multiplier). These are the levers that move exponentials around. The relationship to the common log is ln x = 2.303 log₁₀ x; physical-chemistry problems that quote pH or pK often use log₁₀, while radioactivity and capacitor problems use ln.

The "characteristic time" or time constant deserves a mention. For decay N = N₀ e^(−t/τ), τ is the time after which N has fallen to 1/e ≈ 37% of its initial value. For RC circuits τ = RC; for radioactivity τ = 1/λ. The half-life is related but distinct: T₁/₂ = τ ln 2 ≈ 0.693 τ. NEET routinely asks students to convert between τ and T₁/₂, and to read time constants off the slope of a ln N versus t plot.