Core answer: the dot product a·b=x₁x₂+y₁y₂ is zero exactly when the vectors are perpendicular; matrix multiplication takes left-side rows against right-side columns and is not commutative (AB generally differs from BA); complex multiplication gives (a+bi)(c+di)=(ac−bd)+(ad+bc)i, and division works by rationalizing with the conjugate.
One cheat sheet for three objects
| Object | Notation | Core operations | Classic slip |
|---|---|---|---|
| Vector | (x, y) or (x, y, z) | dot, cross, magnitude | dot product is a scalar |
| Matrix | m×n grid | add (same shape), multiply (inner dims match) | no commutative law |
| Complex | a+bi | four ops, modulus, conjugate | i²=−1, not √1 |
Vectors: dot product and projection
Dot product a·b = x₁x₂ + y₁y₂ = |a||b|cosθ. Two uses: judge the angle (positive = acute, zero = perpendicular, negative = obtuse) and compute projections.
Projection of b onto a = a·b ÷ |a| — force decomposition and lighting falloff both rely on it.
Magnitude |a| = √(x²+y²), extended with z² in 3D.
Matrices: product and determinant
Multiplication: A(m×k) × B(k×n) = C(m×n) — the inner dimension k must match. Entry (i,j) of C is row i of A dotted with column j of B.
2×2 determinant: det = ad − bc. det≠0 means invertible, with inverse (1/det)×[[d,−b],[−c,a]]; det=0 means the matrix squashes the plane onto a line — no inverse.
| Operation | Result shape | Example |
|---|---|---|
| 2×2 determinant | scalar | [[1,2],[3,4]] → 4−6 = −2 |
| 2×2 × 2×2 | 2×2 | row-by-column |
| 2×2 × 2×1 vector | 2×1 | rotate/scale coordinates |
Complex arithmetic and modulus
- Add/subtract: real and imaginary parts separately
- Multiply: (a+bi)(c+di) = (ac−bd) + (ad+bc)i (substituting i²=−1)
- Divide: multiply top and bottom by the conjugate (c−di); the denominator becomes c²+d²
- Modulus: |a+bi| = √(a²+b²) — distance from the origin in the complex plane
Example: splitting gravity on an incline
A 30° incline, weight G=(0, −98)N pointing straight down. Unit vector along the slope: u=(cos30°, −sin30°)=(0.866, −0.5).
Downhill component = G·u = 0×0.866 + (−98)×(−0.5) = 49N — the force pulling the block down; the normal component presses about 84.9N into the slope. Steeper angle θ means a bigger G·sinθ downhill pull.
Example: AC impedance in series
Resistor R=30Ω in series with reactance X=40Ω gives complex impedance Z = 30 + 40i Ω.
Total magnitude |Z| = √(30²+40²) = 50Ω, so at 220V the current is 220÷50 = 4.4A. For parallel branches, Z = Z₁Z₂÷(Z₁+Z₂) — each step is conjugate rationalization.
Common mistakes
- “Matrix products commute”: AB≠BA is the norm. Rotate-then-scale differs from scale-then-rotate.
- “The dot product is a vector”: it is a scalar. The cross product (3D only) is the one returning a vector perpendicular to both.
- Applying √a·√b=√(ab) to negatives: √−1·√−1 ≠ √1 — that is precisely why i²=−1 is a definition, not arithmetic.
- “det=0 only blocks the inverse”: it also means Ax=b has no solution or infinitely many — the matrix collapses a dimension.