NCERT Mathematics (Class 11–12) Cheatsheet

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Sets, Relations & Functions

CLASS 11

Sets (Operations & Laws)

Union (A∪B)Elements in A or B or both

Intersection (A∩B)Elements common to both A and B

Difference (A-B)Elements in A but not in B

Complement (A')U - A (universal set minus A)

De Morgan(A∪B)' = A'∩B' and (A∩B)' = A'∪B'

CommutativeA∪B = B∪A, A∩B = B∩A

Associative(A∪B)∪C = A∪(B∪C)

DistributiveA∪(B∩C) = (A∪B)∩(A∪C)

n(A∪B) = n(A)+n(B)-n(A∩B)

n(A∪B∪C) = n(A)+n(B)+n(C)-n(A∩B)-n(B∩C)-n(C∩A)+n(A∩B∩C)

Power SetSet of all subsets; n(P(A)) = 2ⁿ

Relations & Functions

Type	Definition	Example
One-one (Injective)	Each element maps to unique element	f(x)=2x+1
Onto (Surjective)	Every element of codomain has pre-image	f:R→[-1,1], f(x)=sin(x)
Bijective	Both injective and surjective	f(x)=x, x↦2x is bijection on R⁺
Identity	f(a)=a for all a	I(x)=x
Constant	f(x)=c for all x	f(x)=5
Inverse	f(f⁻¹(x))=x	sin(arcsin(x))=x

Important Functions

Modulusf(x)=|x|, domain=R, range=[0,∞)

Signumsgn(x)={-1,0,1}, greatest integer function

Greatest Integer⌊x⌋ = greatest integer ≤ x

Exponentialeˣ, f(x)=aˣ (a>0), f(0)=1

Logarithmiclogₐx, domain=(0,∞), log(ab)=log(a)+log(b)

Polynomialf(x)=aₙxⁿ+...+a₁x+a₀

Rationalf(x)=p(x)/q(x), q(x)≠0

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Trigonometry

CLASS 11

Standard Values

Angle	sin	cos	tan	cosec	sec	cot
0°	0	1	0	∞	1	∞
30°	1/2	√3/2	1/√3	2	2/√3	√3
45°	1/√2	1/√2	1	√2	√2	1
60°	√3/2	1/2	√3	2/√3	2	1/√3
90°	1	0	∞	1	∞	0

Core Identities

Text

sin²θ + cos²θ = 1          1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ        sin(2θ) = 2 sinθ cosθ
cos(2θ) = cos²θ - sin²θ     cos(2θ) = 2cos²θ - 1 = 1 - 2sin²θ
sin(3θ) = 3sinθ - 4sin³θ     cos(3θ) = 4cos³θ - 3cosθ
tan(3θ) = (3t - t³)/(1-3t²)    where t = tanθ

Sum-to-Product & Product-to-Sum

Text

sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2)
sin A - sin B = 2 cos((A+B)/2) sin((A-B)/2)
cos A + cos B = 2 cos((A+B)/2) cos((A-B)/2)
cos A - cos B = -2 sin((A+B)/2) sin((A-B)/2)
2 sin A cos B = sin(A+B) + sin(A-B)
2 cos A cos B = cos(A+B) + cos(A-B)
2 sin A sin B = cos(A-B) - cos(A+B)
cos²A - sin²A = cos 2A

Inverse Trigonometry

Function	Domain	Range	Principal Value
sin⁻¹	[-1,1]	[-π/2, π/2]	[-π/2, π/2]
cos⁻¹	[-1,1]	[0, π]	[0, π]
tan⁻¹	R	(-π/2, π/2)	(-π/2, π/2)
cot⁻¹	R	(0, π)	(0, π)
sec⁻¹	(-∞,-1]∪[1,∞)	[0,π]∪(π/2,3π/2)	[0,π]∪(π/2,3π/2)
cosec⁻¹	(-∞,-1]∪[1,∞)	[-π/2,π/2]∪(0,π)	[−π/2, π/2]∪(0,π)

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Algebra

CLASS 11

Complex Numbers

Standard Formz = a + ib, i² = -1

Modulus|z| = √(a²+b²)

Conjugatez̄ = a - ib, z·z̄ = |z|² = a²+b²

Polar Formz = r(cosθ + i sinθ) = r·e^(iθ)

Euler's Formulae^(iθ) = cosθ + i sinθ; e^(iπ) + 1 = 0

De Moivre[r(cosθ+i sinθ)]ⁿ = rⁿ(cos nθ + i sin nθ)

Cube Roots of Unity1, ω=(-1+√3i)/2, ω²=(-1-√3i)/2; 1+ω+ω²=0

Quadratic Formulax = (-b±√(b²-4ac)) / 2a

Permutations & Combinations

nPrn!/(n-r)!

nCrn! / (r!(n-r)!)

Circular Perm(n-1)!

nCr = nC(n-r)Symmetry property

SumⁿΣᵣ₌₌ nCr = 2ⁿ

ⁿΣᵣ₌₌ r·nCr = n·2ⁿ⁻¹

Binomial Theorem

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(a + b)ⁿ = ⁿC₀ aⁿ + ⁿC₁ aⁿ⁻¹b + ⁿC₂ aⁿ⁻²b² + ... + ⁿCₙ bⁿ
General term: Tᵣ₊₌ = ⁿCᵣ · aⁿ⁻ᵣ · bʳ
Middle term: when n is even, term = (n/2+1)th
Coefficient of middle term = ⁿCₙ/2
Sum of coefficients = 2ⁿ

Sequences & Series

AP nth termaₙ = a + (n-1)d

AP sumSₙ = n/2 [2a + (n-1)d]

GP nth termaₙ = a·rⁿ⁻¹

GP sum (|r|<1)Sₙ = a(1-rⁿ)/(1-r)

GP sum (r=1)Sₙ = n·a

GP infinite sumS∞ = a/(1-r) when |r|<1

ⁿΣ1 = n

ⁿΣn = n(n+1)/2

ⁿΣn² = n(n+1)(2n+1)/6

ⁿΣn³ = [n(n+1)/2]²

Matrices & Determinants

Text

Matrix Operations: A+B, A-B, scalar multiplication (kA), AB (columns of A × rows of B)
Transpose: (AB)ᵀ = BᵀAᵀ
Symmetric: A = Aᵀ (aᵢⱼ = aⱼⱼ)
Skew-symmetric: A = -Aᵀ (diagonal elements = 0)
|AB| = |A|·|B|    (only if both are square)
A⁻¹ = adj(A)/|A|    (exists only if |A| ≠ 0)
Cramer's Rule: x = |Aₓ|/|A|, y = |Aᵧ|/|A|, z = |Aᶜ|/|A|

Cofactors & Adjoint: Cᵢⱼ = (-1)^(i+j) Mⱼⱼ
Determinant expansion:  |A| = a(ei-fh) - b(di-fg) + c(dh-eg)  (for 3×3)

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Coordinate Geometry

CLASS 11

Key Formulas

Distanced = √[(x₂-x₁)² + (y₂-y₁)²]

Section Formula (internal)P(x,y) divides AB in ratio m:n → ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))

Section Formula (external)P divides AB externally → ((mx₂-nx₁)/(m-n), (my₂-ny₁)/(m-n))

MidpointM = ((x₁+x₂)/2, (y₁+y₂)/2)

Area of TriangleΔ = ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|

CentroidG = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)

Collinearityx₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂) = 0

Straight Line Forms

Form	Equation
Slope-intercept	y = mx + c
Point-slope	y - y₁ = m(x - x₁)
Two-point	(y - y₁)/(y₂-y₁) = (x - x₁)/(x₂-x₁)
Intercept form	x/a + y/b = 1
Normal form	x cosα + y sinα = p
General form	Ax + By + C = 0; slope = -A/B
Distance from point (x₀,y₀) to line Ax+By+C=0	\|Ax₀+By₀+C\|/√(A²+B²)\|

Conic Sections

Conic	Standard Equation	Key Properties
Circle	(x-h)²+(y-k)²=r²	Center(h,k), radius r, area πr²
Parabola	y²=4ax (right), x²=4ay (up)	Focus(a,0), Directrix x=-a, Latus rectum=4a
Ellipse	x²/a²+y²/b²=1	Foci(±ae,0), e=c/a<1, Latus rectum=2b²/a
Hyperbola	x²/a²-y²/b²=1	Foci(±ae,0), e=c/a>1, Asymptotes y=±(b/a)x

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Calculus

CLASS 11-12

Limits — Important Results

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lim(x→0) sin x/x = 1                    lim(x→0) (eˣ-1)/x = 1
lim(x→0) (tan⁻¹x)/x = 1                lim(x→0) (aˣ-1)/x = ln(a)
lim(x→0) (1-cos x)/x = 0                  lim(n→∞) (1+1/n)ⁿ = e
L'Hôpital's Rule: lim f(x)/g(x) = lim f'(x)/g'(x) when 0/0 or ∞/∞ form

Derivative Rules

Text

d/dx (xⁿ) = nxⁿ⁻¹                     d/dx (constant) = 0
d/dx (sin x) = cos x                        d/dx (cos x) = -sin x
d/dx (tan x) = sec²x                       d/dx (cot x) = -cosec²x
d/dx (sec x) = sec x tan x                   d/dx (cosec x) = -cosec x cot x
d/dx (eˣ) = eˣ                            d/dx (aˣ) = aˣ ln(a)
d/dx (ln x) = 1/x                          d/dx (logₐx) = 1/(x ln(a))
Chain Rule: d/dx [f(g(x))] = f'(g(x))·g'(x)
Product Rule: (uv)' = u'v + uv'
Quotient Rule: (u/v)' = (u'v - uv')/v²

Applications of Derivatives

Increasingf'(x) > 0

Decreasingf'(x) < 0

Max/Minf'(x) = 0, check f''(x): +ve→min, -ve→max

Tangenty-y₁ = f'(x₁)(x-x₁)

Normaly-y₁ = -1/f'(x₁)·(x-x₁)

Rate of Changedy/dx = (dy/dt)/(dx/dt)

Rolle'sf(a)=f(b) → ∃c∈(a,b) s.t. f'(c)=0

MVTf'(c) = [f(b)-f(a)]/(b-a)

Lagrange'sf(b)-f(a) = f'(c)(b-a) generalized to g(x)

Integration

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∫ xⁿ dx = xⁿ⁺¹/(n+1) + C           ∫ 1/x dx = ln|x| + C
∫ eˣ dx = eˣ + C                        ∫ sin x dx = -cos x + C
∫ cos x dx = sin x + C                     ∫ sec²x dx = tan x + C
∫ cosec²x dx = -cot x + C                ∫ sec x tan x dx = sec x + C
∫ cosec x cot x dx = -cosec x + C        ∫ aˣ dx = aˣ/ln(a) + C
∫ uv dx = u∫v dx - ∫ u'(∫v dx) dx  (Integration by parts)
∫ eˣⁿ dx = (eˣⁿ/ln(e))·[xⁿ - n·∫ xⁿ⁻¹ dx]  (reduction formula)
Definite: ∫ₐᵇ f(x)dx = F(b) - F(a)
Area = ∫ₐᵇ |y| dx for region between curve and x-axis

Differential Equations

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Variable Separable: dy/dx = f(x)g(y) → ∫dy/g(y) = ∫f(x)dx
Homogeneous: dy/dx = f(y/x) → let y = vx, dy/dx = v + x(dv/dx)
Linear: dy/dx + Py = Q(x)
  IF: IF = e^(∫P dx), solution: y·IF = ∫ Q·IF dx + C
Second Order: a d²y/dx² + b dy/dx + cy = f(x)
  AE = x² + 2mx + n → y = complementary function + particular integral

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Vectors & 3D Geometry

CLASS 12

Vector Operations

Text

Addition: a + b (parallelogram law)
Scalar/Dot Product: a·b = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃
|a·b| = 0 ⟺ a ⊥ b
Vector/Cross Product: |a×b| = |a||b|sinθ, direction: ⊥ to both a and b
(a×b)·(a×b) = |a×b|² = |a|²|b|² - (a·b)²
Scalar Triple Product: [a b c] = a·(b×c) = volume of parallelepiped
    [a b c] = 0 ⟺ coplanar vectors

3D Geometry

Liner = a + λb (parametric)

Angle between linescosθ = (a₁b₁+a₂b₂+a₃b₃)/(|a||b|)

Planer·n̂ = d (normal form)

Shortest distance between skew linesd = |[a₂-a₁ b₂-b₁ c₂-c₁]|/|b₁×b₂|

Distance from point to planed = |Ax₀+By₀+Cz₀+D|/√(A²+B²+C²)

Direction Cosinesl=cosα, m=cosβ, n=cosγ; l²+m²+n²=1

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Probability

CLASS 12

Key Concepts

P(A|B)P(A∩B)/P(B) — probability of A given B occurred

P(A∩B)P(A)·P(B|A) = P(B)·P(A|B)

P(A∪B)P(A)+P(B)-P(A∩B)

Total ProbabilityP(E) = Σ P(Eᵢ)·P(E|Eᵢ)

Bayes'P(Eᵢ|F) = P(F|Eᵢ)·P(Eᵢ)/P(F)

Mean of Binomialμ = np

Variance of Binomialσ² = npq

Std Deviationσ = √(npq)

BernoulliSpecial case of Binomial where n=1; P(X=r) = ᶜCᵣpʳ(1-p)ⁿ⁻ʳ

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Remember: P(A|B) ≠ P(B|A). Bayes' theorem is the formula that "reverses" conditional probability. The binomial distribution models the number of successes in n independent Bernoulli trials.

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Formula Sheet

QUICK REVISION

Quick Reference Formulas

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ALGEBRA
(a+b)² = a² + 2ab + b²             (a-b)² = a² - 2ab + b²
(a+b)³ = a³ + 3a²b + 3ab² + b³     (a-b)³ = a³ - 3a²b + 3ab² - b³
a³ + b³ = (a+b)(a²-ab+b²)         a³ - b³ = (a-b)(a²+ab+b²)
a² - b² = (a+b)(a-b)                x² - y² = (x+y)(x-y)

CALCULUS
d/dx [u/v] = (u'v - uv')/v²              ∫₀ᵇ f(x)dx = F(b)-F(a)
Integration by parts: ∫u dv = uv - ∫v' du
Area under curve = ∫ₐᵇ y dx | Area between curves = ∫ₐᵇ (f-g) dx

TRIGONOMETRY
sin 2A = 2sinA cosA      cos 2A = 2cos²A-1 = 1-2sin²A
tan 2A = 2tanA/(1-tan²A)   3A: sin3A = 3sinA-4sin³A, cos3A = 4cos³A-3cosA

VECTORS
|a+b|² = |a|²+|b|²+2(a·b)    |a×b|² = |a|²|b|²-(a·b)²
[a b c] = a·(b×c)             (a×b) = -(b×a)

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Type

Definition

Example

One-one (Injective)

Each element maps to unique element

f(x)=2x+1

Onto (Surjective)

Every element of codomain has pre-image

f:R→[-1,1], f(x)=sin(x)

Bijective

Both injective and surjective

f(x)=x, x↦2x is bijection on R⁺

Identity

f(a)=a for all a

I(x)=x

Constant

f(x)=c for all x

f(x)=5

Inverse

f(f⁻¹(x))=x

sin(arcsin(x))=x

Angle

sin

cos

tan

cosec

sec

cot

0°

∞

30°

1/2

√3/2

1/√3

2/√3

√3

45°

1/√2

√2

60°

√3/2

1/2

√3

2/√3

1/√3

90°

∞

sin²θ + cos²θ = 1 1 + tan²θ = sec²θ 1 + cot²θ = cosec²θ sin(2θ) = 2 sinθ cosθ cos(2θ) = cos²θ - sin²θ cos(2θ) = 2cos²θ - 1 = 1 - 2sin²θ sin(3θ) = 3sinθ - 4sin³θ cos(3θ) = 4cos³θ - 3cosθ tan(3θ) = (3t - t³)/(1-3t²) where t = tanθ

sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2) sin A - sin B = 2 cos((A+B)/2) sin((A-B)/2) cos A + cos B = 2 cos((A+B)/2) cos((A-B)/2) cos A - cos B = -2 sin((A+B)/2) sin((A-B)/2) 2 sin A cos B = sin(A+B) + sin(A-B) 2 cos A cos B = cos(A+B) + cos(A-B) 2 sin A sin B = cos(A-B) - cos(A+B) cos²A - sin²A = cos 2A

Function

Domain

Range

Principal Value

sin⁻¹

[-1,1]

[-π/2, π/2]

cos⁻¹

[-1,1]

[0, π]

tan⁻¹

(-π/2, π/2)

cot⁻¹

(0, π)

sec⁻¹

(-∞,-1]∪[1,∞)

[0,π]∪(π/2,3π/2)

cosec⁻¹

(-∞,-1]∪[1,∞)

[-π/2,π/2]∪(0,π)

[−π/2, π/2]∪(0,π)

(a + b)ⁿ = ⁿC₀ aⁿ + ⁿC₁ aⁿ⁻¹b + ⁿC₂ aⁿ⁻²b² + ... + ⁿCₙ bⁿ General term: Tᵣ₊₌ = ⁿCᵣ · aⁿ⁻ᵣ · bʳ Middle term: when n is even, term = (n/2+1)th Coefficient of middle term = ⁿCₙ/2 Sum of coefficients = 2ⁿ

Matrix Operations: A+B, A-B, scalar multiplication (kA), AB (columns of A × rows of B) Transpose: (AB)ᵀ = BᵀAᵀ Symmetric: A = Aᵀ (aᵢⱼ = aⱼⱼ) Skew-symmetric: A = -Aᵀ (diagonal elements = 0) |AB| = |A|·|B| (only if both are square) A⁻¹ = adj(A)/|A| (exists only if |A| ≠ 0) Cramer's Rule: x = |Aₓ|/|A|, y = |Aᵧ|/|A|, z = |Aᶜ|/|A| Cofactors & Adjoint: Cᵢⱼ = (-1)^(i+j) Mⱼⱼ Determinant expansion: |A| = a(ei-fh) - b(di-fg) + c(dh-eg) (for 3×3)

Form

Equation

Slope-intercept

y = mx + c

Point-slope

y - y₁ = m(x - x₁)

Two-point

(y - y₁)/(y₂-y₁) = (x - x₁)/(x₂-x₁)

Intercept form

x/a + y/b = 1

Normal form

x cosα + y sinα = p

General form

Ax + By + C = 0; slope = -A/B

Distance from point (x₀,y₀) to line Ax+By+C=0

|Ax₀+By₀+C|/√(A²+B²)|

Conic

Standard Equation

Key Properties

Circle

(x-h)²+(y-k)²=r²

Center(h,k), radius r, area πr²

Parabola

y²=4ax (right), x²=4ay (up)

Focus(a,0), Directrix x=-a, Latus rectum=4a

Ellipse

x²/a²+y²/b²=1

Foci(±ae,0), e=c/a<1, Latus rectum=2b²/a

Hyperbola

x²/a²-y²/b²=1

Foci(±ae,0), e=c/a>1, Asymptotes y=±(b/a)x

lim(x→0) sin x/x = 1 lim(x→0) (eˣ-1)/x = 1 lim(x→0) (tan⁻¹x)/x = 1 lim(x→0) (aˣ-1)/x = ln(a) lim(x→0) (1-cos x)/x = 0 lim(n→∞) (1+1/n)ⁿ = e L'Hôpital's Rule: lim f(x)/g(x) = lim f'(x)/g'(x) when 0/0 or ∞/∞ form

d/dx (xⁿ) = nxⁿ⁻¹ d/dx (constant) = 0 d/dx (sin x) = cos x d/dx (cos x) = -sin x d/dx (tan x) = sec²x d/dx (cot x) = -cosec²x d/dx (sec x) = sec x tan x d/dx (cosec x) = -cosec x cot x d/dx (eˣ) = eˣ d/dx (aˣ) = aˣ ln(a) d/dx (ln x) = 1/x d/dx (logₐx) = 1/(x ln(a)) Chain Rule: d/dx [f(g(x))] = f'(g(x))·g'(x) Product Rule: (uv)' = u'v + uv' Quotient Rule: (u/v)' = (u'v - uv')/v²

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C ∫ 1/x dx = ln|x| + C ∫ eˣ dx = eˣ + C ∫ sin x dx = -cos x + C ∫ cos x dx = sin x + C ∫ sec²x dx = tan x + C ∫ cosec²x dx = -cot x + C ∫ sec x tan x dx = sec x + C ∫ cosec x cot x dx = -cosec x + C ∫ aˣ dx = aˣ/ln(a) + C ∫ uv dx = u∫v dx - ∫ u'(∫v dx) dx (Integration by parts) ∫ eˣⁿ dx = (eˣⁿ/ln(e))·[xⁿ - n·∫ xⁿ⁻¹ dx] (reduction formula) Definite: ∫ₐᵇ f(x)dx = F(b) - F(a) Area = ∫ₐᵇ |y| dx for region between curve and x-axis

Variable Separable: dy/dx = f(x)g(y) → ∫dy/g(y) = ∫f(x)dx Homogeneous: dy/dx = f(y/x) → let y = vx, dy/dx = v + x(dv/dx) Linear: dy/dx + Py = Q(x) IF: IF = e^(∫P dx), solution: y·IF = ∫ Q·IF dx + C Second Order: a d²y/dx² + b dy/dx + cy = f(x) AE = x² + 2mx + n → y = complementary function + particular integral

Addition: a + b (parallelogram law) Scalar/Dot Product: a·b = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃ |a·b| = 0 ⟺ a ⊥ b Vector/Cross Product: |a×b| = |a||b|sinθ, direction: ⊥ to both a and b (a×b)·(a×b) = |a×b|² = |a|²|b|² - (a·b)² Scalar Triple Product: [a b c] = a·(b×c) = volume of parallelepiped [a b c] = 0 ⟺ coplanar vectors

ALGEBRA (a+b)² = a² + 2ab + b² (a-b)² = a² - 2ab + b² (a+b)³ = a³ + 3a²b + 3ab² + b³ (a-b)³ = a³ - 3a²b + 3ab² - b³ a³ + b³ = (a+b)(a²-ab+b²) a³ - b³ = (a-b)(a²+ab+b²) a² - b² = (a+b)(a-b) x² - y² = (x+y)(x-y) CALCULUS d/dx [u/v] = (u'v - uv')/v² ∫₀ᵇ f(x)dx = F(b)-F(a) Integration by parts: ∫u dv = uv - ∫v' du Area under curve = ∫ₐᵇ y dx | Area between curves = ∫ₐᵇ (f-g) dx TRIGONOMETRY sin 2A = 2sinA cosA cos 2A = 2cos²A-1 = 1-2sin²A tan 2A = 2tanA/(1-tan²A) 3A: sin3A = 3sinA-4sin³A, cos3A = 4cos³A-3cosA VECTORS |a+b|² = |a|²+|b|²+2(a·b) |a×b|² = |a|²|b|²-(a·b)² [a b c] = a·(b×c) (a×b) = -(b×a)