What a vector really is — not just a column of numbers, but a geometric object with direction and magnitude. We build dot products from scratch, discover why projection is everywhere, and see how all of this runs through circuits and EM fields.
Forget the definition for a second. You're standing in a field. You walk 3 steps east and 4 steps north. That walk — the direction and how far — is a vector. It doesn't care where you started. It's pure displacement.
You can represent that displacement as an arrow, a column of numbers, or a magnitude + angle. All three are the same object wearing different clothes.
A vector v in 2D lives in ℝ². We write it as a linear combination of the standard basis vectors — the atomic unit arrows along each axis:
The magnitude (or norm) comes straight from Pythagoras — you're measuring the length of the hypotenuse formed by the components.
The generalization doesn't change anything structurally — it's still Pythagoras, just in a higher-dimensional right triangle.
The dot product takes two vectors and produces a single number (a scalar). Two equivalent definitions — one algebraic, one geometric — that always produce the same answer.
The dot product measures how much two vectors point in the same direction. cos(θ) = 1 when perfectly aligned, 0 when perpendicular, −1 when opposite. The visualization below shows both formulas updating simultaneously — they always agree.
Let's derive a·b = ‖a‖‖b‖cos(θ) from the algebraic definition. Do this once and you'll never forget the connection between algebra and geometry.
The algebra and the geometry aren't two separate definitions — they're the same truth seen from two angles. This is the Feynman move: when two formulas always give the same number, they're revealing something deep about the structure of space.
Projection answers: "how much of a points in the direction of b?" Geometrically, it's the shadow of a cast onto b. This single operation is behind AC power, Gauss's law, least squares, and Fourier series.
Read the last line: find how much of a aligns with b̂ (that's the dot product — a scalar), then scale b̂ by that scalar to get the shadow vector. Dot with a direction → how far you go in that direction.
In 3D, vectors have three components. All the same rules hold — just one more term in the sums. The intuition is identical:
In ℝⁿ for any n? Same thing. The intuition you built in 2D extends perfectly to 1000 dimensions — you can't visualize it, but the algebra carries the geometry for you.
Real power in AC circuits. In phasor form, voltage and current are vectors. Real power (watts) = V · I = ‖V‖‖I‖cos(φ) — the projection of current onto voltage. The perpendicular component (reactive power, VAR) does no work. Power factor is literally cos(θ).
Electric flux. Φ = E · A = ‖E‖‖A‖cos(θ). If the field is parallel to the surface (θ = 90°), flux is zero. If perpendicular (θ = 0°), flux is maximum. The dot product encodes exactly how much field "passes through" the surface.
Rate of change of a scalar field. The derivative of field φ in direction d̂ is ∇φ · d̂. Pure projection. The gradient points in the direction where this dot product is maximized — where cos(θ) = 1.
a · b = 0 ⟺ a ⊥ b. Two vectors are perpendicular exactly when their dot product is zero, because cos(90°) = 0. This single fact is why Fourier series work, why least squares works, and why basis vectors are chosen perpendicular. It is the most important geometric fact in all of linear algebra.