CELE Engineering Mathematics — Advanced Engineering MathematicsCheat Sheet

Formula

z = a + bi

Meaning

Rectangular form: a = real part, b = imaginary part, i = √(−1)

Watch Out

Students forget i² = −1; always simplify imaginary products immediately.

When To Use

Starting point for most complex number problems; converting from polar or exponential forms.

Formula

|z| = √(a² + b²)

Meaning

Magnitude (modulus) of z; the distance from origin to z in the complex plane.

Watch Out

Common error: |z| is NEVER negative; always take the positive square root.

When To Use

Always required when converting to polar or exponential form; needed for division/multiplication in polar form.

Formula

θ = arctan(b/a) [adjust for quadrant]

Meaning

Argument (angle); θ is measured counterclockwise from the positive real axis in degrees or radians.

Watch Out

Quadrant errors: Q1 (a>0, b>0) use arctan directly; Q2 (a<0, b>0) add 180°; Q3 (a<0, b<0) add 180°; Q4 (a>0, b<0) add 360° or use negative angle.

When To Use

When converting rectangular to polar form; must check which quadrant a and b place z in.

Formula

z = r∠θ = r(cos θ + i sin θ)

Meaning

Polar form: r = |z| magnitude, θ = argument in degrees or radians.

Watch Out

Ensure θ is in the correct range (0° to 360° or −180° to 180°); radian vs degree consistency.

When To Use

Multiplication, division, powers, and roots are MUCH simpler in polar form.

Formula

z = r·e^(iθ)

Meaning

Exponential (Euler) form: e^(iθ) = cos θ + i sin θ.

Watch Out

Requires θ in RADIANS; not degree mode. Easy to confuse with polar notation.

When To Use

Advanced work with differential equations, signal processing; equivalent to polar form.

Formula

z₁ · z₂ = r₁r₂ ∠(θ₁ + θ₂) [polar form]

Meaning

Multiply magnitudes; add arguments.

Watch Out

Forget to reduce final angle to 0°–360° range; students mix up 'add angles' with 'add real parts'.

When To Use

Always use polar form for multiplication; rectangular form is tedious.

Formula

z₁ ÷ z₂ = (r₁/r₂) ∠(θ₁ − θ₂) [polar form]

Meaning

Divide magnitudes; subtract arguments.

Watch Out

Subtraction of angles can yield negative angles; adjust to 0°–360° if needed.

When To Use

Division is vastly simpler in polar form than rectangular.

Formula

z^n = r^n ∠(nθ) [De Moivre's Theorem]

Meaning

Raise magnitude to power n; multiply argument by n.

Watch Out

The final angle may exceed 360°; always reduce to standard range. Do NOT expand binomially.

When To Use

Powers of complex numbers; infinitely simpler than expanding (a+bi)^n in rectangular form.

Formula

z^(1/n) = r^(1/n) ∠[(θ + 360°k)/n] for k = 0, 1, ..., n−1

Meaning

n distinct nth roots; magnitude root is r^(1/n); arguments are (θ + 360k°)/n for each k.

Watch Out

There are EXACTLY n distinct roots; students often forget to generate all k values. Missing even one root is a loss of marks.

When To Use

Finding square roots, cube roots, etc. of complex numbers.

Formula

(a + bi)(a − bi) = a² + b²

Meaning

Product of a complex number and its conjugate equals the square of its magnitude (real number).

Watch Out

Conjugate of a+bi is a−bi (flip sign of imaginary part). Easy to forget the sign flip.

When To Use

Rationalizing denominators in complex division; always yields a real result.

Value

0°, 30°, 45°, 60°, 90°, 180°, 270°

Symbol

θ

Quantity

Common argument angles

Value

sin 30° = 0.5, cos 30° = √3/2; sin 45° = √2/2, cos 45° = √2/2; sin 60° = √3/2, cos 60° = 0.5; sin 90° = 1, cos 90° = 0

Symbol

sin θ, cos θ

Quantity

sin and cos at standard angles

Value

1 radian ≈ 57.3°; 1° ≈ 0.0175 rad

Symbol

rad ↔ °

Quantity

Conversion factor

Term

Complex number

Example

3 + 4i (real part 3, imaginary part 4)

Definition

Number of the form a + bi where a, b are real and i² = −1.

Term

Magnitude / Modulus

Example

|3 + 4i| = √(9 + 16) = 5

Definition

Distance from origin to point z in complex plane; |z| = √(a² + b²).

Term

Argument

Example

arg(3 + 4i) ≈ 53.13°

Definition

Angle from positive real axis to point z, measured counterclockwise; typically 0° to 360° or −180° to 180°.

Term

Conjugate

Example

Conjugate of 2 + 3i is 2 − 3i

Definition

Complex number with imaginary part negated; conjugate of a+bi is a−bi.

Term

Polar form

Example

3 + 4i = 5∠53.13°

Definition

Representation as r∠θ or r(cos θ + i sin θ); used for multiplication, division, powers.

Formula

det(2×2) = |a b| = ad − bc |c d|

Meaning

For a 2×2 matrix, determinant = (product of main diagonal) − (product of anti-diagonal).

Watch Out

Order matters: ad − bc, NOT ad + bc. Sign errors are the #1 mistake.

When To Use

Calculating determinant for 2×2 systems; foundation for larger matrices and Cramer's rule.

Formula

det(3×3) = a(ei−fh) − b(di−fg) + c(dh−eg) [cofactor expansion along first row]

Meaning

For 3×3, expand along any row or column using cofactors (−1)^(i+j) times the minor.

Watch Out

Sign pattern alternates: +, −, +, − along expansions. Easy to flip a sign; always double-check cofactor signs.

When To Use

Solving 3×3 systems with Cramer's rule; finding matrix inverses.

Formula

A^(−1) = (1/det A) × adj(A)

Meaning

For invertible square matrix A, inverse = (reciprocal of determinant) × (adjugate matrix).

Watch Out

If det A = 0, matrix is SINGULAR; no inverse exists. Adjugate is transpose of cofactor matrix.

When To Use

Solving Ax = b as x = A^(−1)b; required when det A ≠ 0.

Formula

A^(−1) = (1/(ad−bc)) × |d −b| [2×2 specific] |−c a|

Meaning

For 2×2: swap a↔d on diagonal, negate b and c off-diagonal, divide by determinant.

Watch Out

Must verify det(A) ≠ 0 FIRST. The swap-and-negate pattern is easy to mess up.

When To Use

Quick inversion of 2×2 matrices; much faster than cofactor method for small systems.

Formula

A · B ≠ B · A [matrix multiplication is NOT commutative]

Meaning

Order of multiplication matters; result depends on which matrix is on the left.

Watch Out

Treating matrices like scalars and assuming AB = BA is a critical error in exams.

When To Use

Any matrix algebra problem; reminds you to keep track of order.

Formula

(m × n) · (n × p) = (m × p)

Meaning

Inner dimensions must match; result has outer dimensions.

Watch Out

If inner dimensions don't match, multiplication is UNDEFINED. Always verify dimensions first.

When To Use

Checking if two matrices can be multiplied; determining size of product.

Formula

det(A·B) = det(A) · det(B)

Meaning

Determinant of a product equals the product of determinants.

Watch Out

This is a theorem; NOT true for det(A+B). Do NOT confuse with additive property.

When To Use

Simplifying determinant calculations; checking invertibility of products.

Formula

det(A^T) = det(A) [transpose property]

Meaning

Determinant of a transposed matrix equals the original determinant.

Watch Out

Transpose does NOT change the determinant value; keep this in mind for row/column operations.

When To Use

Simplifying problems where you encounter A^T; often used in proofs.

Value

[1 0] [0 1]

Symbol

I₂

Quantity

Identity matrix (2×2)

Value

[1 0 0] [0 1 0] [0 0 1]

Symbol

I₃

Quantity

Identity matrix (3×3)

Term

Matrix

Example

[1 2 3] [4 5 6] is a 2×3 matrix

Definition

Rectangular array of numbers arranged in rows and columns; denoted m × n (m rows, n columns).

Term

Determinant

Example

det([2 3] [1 4]) = 2(4) − 3(1) = 5

Definition

Single scalar value computed from a square matrix; indicates invertibility and scaling effect.

Term

Inverse matrix

Example

For A = [1 2], A^(−1) = [4 −2] [3 4] [−3 1] (after dividing by det A = −2)

Definition

Matrix A^(−1) such that A · A^(−1) = A^(−1) · A = I (identity); exists only if det A ≠ 0.

Term

Minor

Example

For 3×3 matrix, minor M₁₂ is the determinant after deleting row 1 and column 2

Definition

Determinant of the matrix formed by deleting one row and one column.

Term

Cofactor

Example

C₁₁ = (−1)^(1+1) M₁₁ = +M₁₁; C₁₂ = (−1)^(1+2) M₁₂ = −M₁₂

Definition

Minor with a sign: C_ij = (−1)^(i+j) · M_ij.

Term

Adjugate matrix

Example

If cofactor matrix is C, then adj(A) = C^T

Definition

Transpose of the cofactor matrix; denoted adj(A).

Term

Singular matrix

Example

Any matrix with a row of all zeros or two identical rows is singular

Definition

Square matrix with det = 0; not invertible; rows or columns are linearly dependent.

Formula

For Ax = b, x_i = (det A_i) / (det A)

Meaning

Each unknown x_i equals the ratio of two determinants: numerator is A with ith column replaced by b, denominator is det(A).

Watch Out

ONLY works if det A ≠ 0. If det A = 0, no unique solution (either no solution or infinite solutions).

When To Use

Solving 2×2 or 3×3 linear systems with unique solution; requires det A ≠ 0.

Formula

x = (det A₁) / (det A), y = (det A₂) / (det A), z = (det A₃) / (det A) [for 3×3]

Meaning

For system of 3 equations in 3 unknowns: A₁ replaces column 1 of A with b, A₂ replaces column 2, A₃ replaces column 3.

Watch Out

Easy to confuse which column gets replaced; always double-check that you're replacing the correct column for each variable.

When To Use

Standard application for 3×3 systems; each unknown requires one determinant calculation.

Formula

det A ≠ 0 ⟹ unique solution

Meaning

Non-zero determinant of coefficient matrix guarantees exactly one solution.

Watch Out

If det A = 0, Cramer's rule fails; the system is either inconsistent or dependent (infinitely many solutions).

When To Use

Before applying Cramer's rule; verify the system is solvable with a unique answer.

Term

Coefficient matrix A

Example

For 2x + 3y = 8 and x + 4y = 9, A = [2 3] [1 4]

Definition

Square matrix of all coefficients from the left side of the system; does not include constants b.

Term

Constant vector b

Example

For the system above, b = [8] [9]

Definition

Column vector of all right-hand-side constants from the equations.

Term

Augmented matrix A_i

Example

For x calculation in 2×2, A₁ = [8 3] [9 4] (column 1 replaced with b)

Definition

Matrix formed by replacing the ith column of A with vector b.

Term

Unique solution

Example

System 2x + 3y = 8, x + 4y = 9 has unique solution x = 1, y = 2

Definition

Exactly one set of values satisfying all equations simultaneously; occurs when det A ≠ 0.

Formula

**a** = (a_x, a_y, a_z) [component form]

Meaning

Vector represented as ordered triple of components along x, y, z axes.

Watch Out

Do NOT confuse vector (a_x, a_y, a_z) with point (a_x, a_y, a_z); vectors have magnitude AND direction.

When To Use

Starting point for all vector calculations; easy to work with componentwise.

Formula

|**a**| = √(a_x² + a_y² + a_z²)

Meaning

Magnitude (length) of vector; distance from origin if **a** is position vector.

Watch Out

Magnitude is ALWAYS non-negative. Magnitude equals zero only if vector is the zero vector (0, 0, 0).

When To Use

Whenever you need the 'length' or 'size' of a vector; required for unit vectors.

Formula

**a** · **b** = a_x·b_x + a_y·b_y + a_z·b_z

Meaning

Dot product (scalar product): sum of products of corresponding components.

Watch Out

Dot product yields a SCALAR (single number), never a vector. If result is 0, vectors are perpendicular.

When To Use

Computing angles between vectors, checking perpendicularity, projections.

Formula

**a** · **b** = |**a**| |**b**| cos θ

Meaning

Geometric interpretation: dot product equals product of magnitudes times cosine of angle between them.

Watch Out

θ is always between 0° and 180° (or 0 and π rad). If dot product is 0, θ = 90° (perpendicular).

When To Use

Finding angle θ between two vectors: cos θ = (**a** · **b**) / (|**a**| |**b**|), then θ = arccos(...).

Formula

**a** × **b** = |**i** **j** **k** | |a_x a_y a_z | |b_x b_y b_z |

Meaning

Cross product (vector product): determinant expansion yielding a vector perpendicular to both **a** and **b**.

Watch Out

Cross product is a VECTOR (three components), NOT a scalar. **a** × **b** ≠ **b** × **a**; order matters (reverses direction).

When To Use

Finding perpendicular vectors, calculating areas of parallelograms, torque calculations.

Formula

**a** × **b** = (a_y·b_z − a_z·b_y, a_z·b_x − a_x·b_z, a_x·b_y − a_y·b_x)

Meaning

Component form of cross product after determinant expansion.

Watch Out

Easy to mix up signs and component order. Double-check the cyclic pattern: (y·z−z·y, z·x−x·z, x·y−y·x).

When To Use

Practical calculation of cross product; often faster than full determinant notation.

Formula

|**a** × **b**| = |**a**| |**b**| sin θ

Meaning

Magnitude of cross product: product of magnitudes times sine of angle.

Watch Out

If **a** × **b** = **0**, then **a** and **b** are PARALLEL (θ = 0° or 180°), not perpendicular.

When To Use

Area of parallelogram spanned by **a** and **b** equals |**a** × **b**|; if sin θ = 0, vectors are parallel.

Formula

**a** × **b** ⊥ **a** and **a** × **b** ⊥ **b** [perpendicularity property]

Meaning

Cross product is always perpendicular to both input vectors.

Watch Out

This is a defining property of cross product; always true for non-parallel vectors.

When To Use

Finding normal vectors to a plane; knowing the resulting vector direction.

Formula

Unit vector: **u** = **a** / |**a**|

Meaning

Vector with magnitude 1 in the same direction as **a**.

Watch Out

Cannot divide by zero; |**a**| must be non-zero. Always check the vector is not the zero vector.

When To Use

Normalizing vectors; expressing direction without magnitude information.

Formula

Projection of **a** onto **b**: proj_**b**(**a**) = ((**a**·**b**) / |**b**|²) **b**

Meaning

Component of **a** in the direction of **b**; a vector parallel to **b**.

Watch Out

Projection is a VECTOR, not a scalar. If dot product is negative, projection points opposite to **b**.

When To Use

Decomposing forces, finding components in specific directions.

Value

**i** = (1, 0, 0); **j** = (0, 1, 0); **k** = (0, 0, 1)

Symbol

**i**, **j**, **k**

Quantity

Standard basis vectors

Value

θ = 90° or π/2 radians

Symbol

θ

Quantity

Angle where cos θ = 0 (perpendicular)

Value

θ = 0° or 180° (0 or π radians)

Symbol

θ

Quantity

Angle where sin θ = 0 (parallel)

Term

Vector

Example

**a** = (3, 4, 0) has magnitude 5 and direction in the xy-plane

Definition

Quantity with both magnitude and direction; represented as ordered list of components or arrow in space.

Term

Dot product (scalar product)

Example

(1, 2, 3) · (4, 5, 6) = 1(4) + 2(5) + 3(6) = 32

Definition

Binary operation yielding a scalar; measures 'alignment' of two vectors.

Term

Cross product (vector product)

Example

(1, 0, 0) × (0, 1, 0) = (0, 0, 1) (right-hand rule applies)

Definition

Binary operation yielding a vector perpendicular to both inputs; non-commutative.

Term

Magnitude

Example

|(3, 4, 0)| = √(9 + 16 + 0) = 5

Definition

Length of a vector; always non-negative; |**a**| = √(a_x² + a_y² + a_z²).

Term

Unit vector

Example

Unit vector in direction of (3, 4, 0) is (3/5, 4/5, 0)

Definition

Vector with magnitude exactly 1; used to represent pure direction.

Term

Angle between vectors

Example

Angle between (1, 0, 0) and (1, 1, 0): cos θ = 1/(1·√2) = 1/√2, so θ = 45°

Definition

Angle θ ∈ [0°, 180°] found using cos θ = (**a** · **b**) / (|**a**| |**b**|).

Term

Perpendicular (orthogonal) vectors

Example

(1, 0, 0) and (0, 1, 0) are perpendicular; their dot product is 0

Definition

Two vectors **a** and **b** are perpendicular if **a** · **b** = 0.

Term

Parallel vectors

Example

(2, 4, 6) and (1, 2, 3) are parallel; (1, 2, 3) = ½(2, 4, 6)

Definition

Two vectors **a** and **b** are parallel if **a** × **b** = **0** (or one is a scalar multiple of the other).

Values

• Scalar (single number)
• Vector (three components)

Property

Result type

Values

• Measure of alignment; |**a**||**b**|cos θ
• Perpendicular vector; magnitude |**a**||**b**|sin θ

Property

Geometric meaning

Values

• **a** · **b** = a_x·b_x + a_y·b_y + a_z·b_z
• **a** × **b** = determinant with **i**, **j**, **k**

Property

Formula

Values

• YES: **a** · **b** = **b** · **a**
• NO: **a** × **b** = −(**b** × **a**)

Property

Commutative?

Values

• Perpendicular vectors (θ = 90°)
• Parallel vectors (θ = 0° or 180°)

Property

Zero result means

Values

• Angles, projections, work, power
• Normal vectors, areas, torque, orientation

Property

When to use

Values

• z = a + bi
• a (real), b (imaginary)
• Addition, subtraction, conjugates

Property

Rectangular

Values

• z = r∠θ
• r (magnitude), θ (argument)
• Multiplication, division, powers, roots

Property

Polar

Values

• z = r·e^(iθ)
• r, θ (in radians)
• Differential equations, advanced applications

Property

Exponential

Values

• Non-singular (invertible)
• Unique solution: x = A^(−1)b

Property

det(A) ≠ 0

Values

• Singular (not invertible)
• No solution or infinite solutions (inconsistent or dependent)

Property

det(A) = 0

Values

• YES
• Efficient and clean; each unknown from one ratio of determinants

Property

det(A) ≠ 0, small system (2×2 or 3×3)

Values

• NO
• Cramer's rule fails; use Gaussian elimination to check consistency

Property

det(A) = 0

Values

• NO
• Computationally expensive; Gaussian elimination is faster

Property

Large system (4×4 or bigger)