Scalar triple product

“If it's zero, the three vectors are coplanar.”

a·(b×c) gives the volume of the parallelepiped formed by three vectors.

[a b c]

Memory trick: Zero → coplanar vectors.

Projection of a vector

“It's the 'shadow' one vector casts along another.”

The projection of a on b is (a·b)/|b|.

(a·b)/|b|

Memory trick: Uses the dot product.

Direction cosines

“They pin down a line's direction in 3-D space.”

The cosines of the angles a line makes with the axes; they satisfy l²+m²+n²=1.

Memory trick: l²+m²+n² = 1.

Angle between two planes

“Two walls meet at the angle their perpendiculars make.”

Equals the angle between their normal vectors.

Memory trick: Compare the normals.

Transpose of a matrix

“Flip a matrix across its diagonal.”

Swap rows and columns to get Aᵀ.

Memory trick: Rows become columns.

Symmetric and skew-symmetric

“Any square matrix splits into a symmetric plus a skew-symmetric part.”

Symmetric: Aᵀ=A; skew-symmetric: Aᵀ=−A (with zero diagonal).

Memory trick: Aᵀ=A vs Aᵀ=−A.

Inverse of a matrix

“The inverse 'undoes' a matrix transformation — if the determinant isn't zero.”

A⁻¹ = adj(A)/det(A), and exists only when det(A) ≠ 0.

A⁻¹=adjA/detA

Memory trick: No inverse when det = 0.

Cramer's rule

“A neat determinant recipe for solving simultaneous equations.”

Solves linear systems using ratios of determinants.

Memory trick: Each unknown = ratio of determinants.

Area of a triangle by determinant

“A determinant of zero means the three points are collinear.”

The area from vertices is half the absolute value of a 3×3 coordinate determinant.

Memory trick: Zero area → collinear points.

Addition theorem of probability

“Subtract the overlap so you don't count it twice.”

P(A∪B) = P(A) + P(B) − P(A∩B).

P(A∪B)=P(A)+P(B)−P(A∩B)

Memory trick: Don't double-count the overlap.

Multiplication theorem

“Two fair coins both landing heads: ½ × ½ = ¼.”

For independent events, P(A∩B) = P(A)·P(B).

Memory trick: Independent → multiply.

Binomial distribution

“It predicts how many heads in 10 coin tosses, or defects in a batch.”

Models the number of successes in n independent yes/no trials with fixed probability p.

Memory trick: n trials, probability p each.

Expected value

“Casinos stay profitable because the expected value tilts their way.”

The long-run average outcome, weighting each value by its probability.

Memory trick: Probability-weighted average.

Linear programming

“Factories use it to maximise profit within limited resources.”

Optimises a linear objective subject to linear constraints; the best value sits at a corner of the feasible region.

Memory trick: Optimum lives at a corner.

Mathematical induction

“Knock the first domino and set up the chain — they all fall.”

Prove a statement for n=1, then that truth at k implies truth at k+1.

Memory trick: Base case + inductive step.

Modulus inequality

“A modulus inequality carves out an interval on the number line.”

|x| < a means −a < x < a; |x| > a means x < −a or x > a.

Memory trick: '<' interval, '>' two rays.

Periodic function

“sin and cos repeat every 2π — the maths of anything cyclic.”

A function that repeats its values at regular intervals (period).

Memory trick: Repeats every period.

Signum function

“It's a switch that reports only the direction, not the size.”

sgn(x) returns −1, 0 or +1 depending on the sign of x.

Memory trick: Reads the sign only.

Product-to-sum formulas

“They turn stubborn products into friendly sums for integration.”

2sinA cosB = sin(A+B) + sin(A−B) and similar identities.

Memory trick: Product → sum.

Maximum of a sinθ + b cosθ

“A quick way to bound any sine-plus-cosine expression.”

It ranges between −√(a²+b²) and +√(a²+b²).

√(a²+b²)

Memory trick: Amplitude = √(a²+b²).

Half-angle formulas

“They let you climb down from an angle to its half.”

sin(θ/2), cos(θ/2) can be written using cosθ, halving the angle.

Memory trick: Express half-angle via cosθ.

nth roots of a complex number

“They sit like the numbers on a clock face around the origin.”

A non-zero complex number has exactly n distinct nth roots, evenly spaced on a circle.

Memory trick: n roots, equally spaced.

Argument of a product

“Multiplying complex numbers adds their angles — a rotation.”

arg(z₁z₂) = arg z₁ + arg z₂.

Memory trick: Multiply → add angles.

Arithmetico-geometric progression

“It blends steady steps with multiplying growth.”

Each term is a product of an AP term and a GP term.

Memory trick: AP × GP terms.