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Class 11 · Chapter 1

Basic Mathematics & Vectors

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Basic Mathematics & Vectors Formula Sheet

16 formulas across 4 topics in Basic Mathematics & Vectors.

2 min read

Updated 2026-07-08 · v1.0.0

Trigonometric & Binomial Approximations

Physics constantly linearises small quantities. These two approximations turn a hard trigonometric or algebraic expression into a manageable linear one — and JEE/NEET reward recognising when they apply.

sinθ ≈ θ, tanθ ≈ θ, cosθ ≈ 1 − θ²/2 (θ in radians, θ → 0)

Small-angle approximation

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Taylor series of sin, cos, tan truncated to leading order

Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
θangle, must be in RADIANSrad[M⁰L⁰T⁰]

Valid when

  • Only valid for θ in radians, and only for small θ (roughly θ < 0.2 rad, or ~10-15°)

Common mistakes

  • Using θ in degrees directly in sinθ ≈ θ — the approximation ONLY holds with θ measured in radians
Read the concept →

(1 + x)ⁿ ≈ 1 + nx (for |x| ≪ 1)

Binomial approximation

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Binomial expansion (1+x)ⁿ = 1 + nx + n(n-1)x²/2! + ⋯, truncated after the linear term

Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
nany real exponent (need not be a positive integer)dimensionless[M⁰L⁰T⁰]
xsmall quantitydimensionless[M⁰L⁰T⁰]

Valid when

  • Requires |x| ≪ 1 — the smaller x, the better the approximation
  • Works for negative and fractional n too, e.g. (1+x)⁻¹ ≈ 1−x, (1+x)^(1/2) ≈ 1 + x/2

Common mistakes

  • Applying it when x is not actually small gives a badly wrong answer — always check the smallness condition before using it
Read the concept →

x = [−b ± √(b² − 4ac)] / 2a for ax² + bx + c = 0

Quadratic formula

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Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
a, b, ccoefficients of the quadratic equationcontext-dependent[context-dependent]

Valid when

  • Discard the root that gives a physically impossible answer (e.g. negative time, negative distance)

Common mistakes

  • Forgetting to reject the unphysical root — both roots are mathematically valid but only one may make physical sense
Read the concept →

Worth remembering

  • Both approximations are 'zoom in and it looks like a straight line' tools — recognising a small parameter in a problem is a hint to reach for one of these

Differentiation & Integration Essentials

Physics defines velocity, acceleration, and countless other quantities as rates of change or accumulated sums — calculus is the language, not an optional extra.

d/dx(xⁿ) = n xⁿ⁻¹

Power rule of differentiation

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Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
nany real exponentdimensionless[M⁰L⁰T⁰]

Valid when

  • The single most-used calculus rule in mechanics — used to get v = dx/dt and a = dv/dt from a position function
Read the concept →

d/dx(sinx) = cosx ; d/dx(cosx) = −sinx ; d/dx(eˣ) = eˣ

Standard derivatives used in physics

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Valid when

  • x must be in radians for the trigonometric derivatives to hold in this simple form
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Maximum: dy/dx = 0 and d²y/dx² < 0 ; Minimum: dy/dx = 0 and d²y/dx² > 0

Condition for maxima and minima

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First derivative gives the stationary point; the sign of the second derivative classifies it

Valid when

  • A stationary point (dy/dx = 0) with d²y/dx² = 0 needs further checking — it may be a point of inflection, not an extremum
Read the concept →

∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)

Power rule of integration

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Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
Cconstant of integrationcontext-dependent[context-dependent]

Valid when

  • Used constantly to reconstruct v from a(t), or x from v(t), given initial conditions to fix C
Read the concept →

Area under y(x) from a to b = ∫[a to b] y dx

Definite integral as area under a curve

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Limit of a Riemann sum of infinitesimally thin strips

Valid when

  • The classic physics use: area under a v–t graph gives displacement; area under F–x graph gives work done

Common mistakes

  • Signed area: regions below the x-axis subtract from the total, they don't automatically add as positive area
Read the concept →

Vector Representation & Addition

A vector needs both magnitude and direction to be fully specified. Addition of vectors is never simple arithmetic — it always accounts for the angle between them.

|A| = √(Ax² + Ay² + Az²)

Magnitude of a vector from its rectangular components

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Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
Ax, Ay, Azcomponents of A along the x, y, z axessame as Asame as A

Valid when

  • Components must be along MUTUALLY PERPENDICULAR axes
Read the concept →

 = A / |A|

Unit vector along a given vector

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Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
Âunit vector, magnitude exactly 1, dimensionlessdimensionless[M⁰L⁰T⁰]

Valid when

  • A unit vector carries direction information only — never carries the physical dimensions of A itself
Read the concept →

R = √(A² + B² + 2AB cosθ)

Magnitude of the resultant of two vectors (parallelogram law)

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Law of cosines applied to the triangle formed by A, B, and their resultant

Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
A, Bmagnitudes of the two vectors being addedcontext-dependentcontext-dependent
θangle between the two vectorsrad[M⁰L⁰T⁰]

Valid when

  • R is maximum (= A+B) when θ = 0°, minimum (= |A−B|) when θ = 180°

Common mistakes

  • The angle θ is the angle BETWEEN the vectors as drawn tail-to-tail, not the angle in the parallelogram diagram's corner in every orientation — draw it carefully
Read the concept →

tanβ = B sinθ / (A + B cosθ)

Direction of the resultant of two vectors

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Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
βangle the resultant makes with vector Arad[M⁰L⁰T⁰]

Valid when

  • β leans toward whichever of A or B has the larger magnitude
Read the concept →

|A − B| = √(A² + B² − 2AB cosθ)

Magnitude of the difference of two vectors

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A − B = A + (−B); apply the addition formula with the angle between A and −B, which is (180° − θ)

Valid when

  • Maximum |A−B| = A+B when θ = 180° (vectors antiparallel); minimum = |A−B| when θ = 0°

Common mistakes

  • Subtraction is NOT commutative in direction: A−B and B−A have the same magnitude but point in opposite directions
Read the concept →

Worth remembering

  • Relative velocity problems are vector subtraction in disguise: v(A relative to B) = v(A) − v(B)

NEET frequently tests the special cases θ = 0°, 90°, 180° for both addition and subtraction as one-line MCQs — memorise the three magnitudes (A+B, √(A²+B²), |A−B|) directly.

Scalar and Vector Products

Multiplying two vectors gives either a scalar (dot product) or another vector (cross product) — physics uses both constantly, and picking the right one is half the battle in a problem.

A·B = AB cosθ = AxBx + AyBy + AzBz

Scalar (dot) product of two vectors

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Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
A·Bdot product, a pure scalarproduct of A and B's unitsproduct of A and B's dimensions

Valid when

  • A·B = 0 when the vectors are perpendicular (θ = 90°) — the basis of the 'no work done' rule when force ⊥ displacement

Common mistakes

  • Dot product is commutative (A·B = B·A), unlike the cross product — a frequent point of confusion
Read the concept →

A × B = AB sinθ n̂ ; |A × B| = AB sinθ

Vector (cross) product of two vectors

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Right-hand rule gives n̂'s direction, perpendicular to the plane containing A and B

Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
n̂unit vector perpendicular to both A and B, direction from the right-hand ruledimensionless[M⁰L⁰T⁰]

Valid when

  • |A × B| = 0 when the vectors are parallel/antiparallel (θ = 0° or 180°)

Common mistakes

  • Cross product is ANTI-commutative: A × B = −(B × A) — swapping the order flips the direction, unlike the dot product
Read the concept →

Area of parallelogram = |A × B| ; Area of triangle = ½|A × B|

Area of a parallelogram/triangle from two adjacent sides

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Geometric meaning of the cross product's magnitude (base × height = AB sinθ)

Valid when

  • A and B must be the two vectors representing adjacent sides emanating from the same vertex
Read the concept →

A recurring JEE Advanced trick: express torque (r × F), angular momentum (r × p), and magnetic force (qv × B) all use the identical cross-product mechanics — practise the determinant method once and reuse it everywhere.

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