Vectors and Dot Product

Summary

Vectors encode magnitude and direction. The dot product measures alignment: it yields lengths, angles, and orthogonal projections, and is the bridge from analytic geometry to linear algebra orthogonality.

Prerequisites

Distance and Midpoint. Helpful: Trigonometry. Hub: Analytic Geometry.

Object / Concept

A vector in $R^{n}$ is an ordered $n$ -tuple $v = ⟨ v_{1}, \dots, v_{n} ⟩$ (equivalently a column). Vectors add componentwise and scale by real numbers.

The dot product (standard inner product) of $u = ⟨ u_{1}, \dots, u_{n} ⟩$ and $v = ⟨ v_{1}, \dots, v_{n} ⟩$ is

u \cdot v = u_{1} v_{1} + \dots + u_{n} v_{n} .

Notation

Symbol	Meaning
$u, v, w$	vectors
$\| v \|$	Euclidean norm (length) of $v$
$u \cdot v$	dot product
${proj}_{v} u$	orthogonal projection of $u$ onto $v$
$0$	zero vector

Conditions / Assumptions

Real Euclidean space $R^{n}$ with the standard dot product.
Projection formula requires $v \neq 0$ .
Angle formula requires $u, v \neq 0$ .

Equations

Length

∥ v ∥ = \sqrt{v \cdot v} = \sqrt{v_{1}^{2} + \dots + v_{n}^{2}} .

Angle $θ$ between nonzero vectors

u \cdot v = ∥ u ∥ ∥ v ∥ \cos θ, \cos θ = \frac{u \cdot v}{∥ u ∥ ∥ v ∥} .

Orthogonality: $u ⊥ v$ if and only if $u \cdot v = 0$ .

Unit vector in the direction of $v \neq 0$ :

\hat{v} = \frac{v}{∥ v ∥} .

Orthogonal projection of $u$ onto $v \neq 0$ :

{proj}_{v} u = (\frac{u \cdot v}{v \cdot v}) v = (\frac{u \cdot v}{∥ v ∥^{2}}) v .

The residual $u - {proj}_{v} u$ is orthogonal to $v$ .

Algebraic properties

u \cdot v = v \cdot u, u \cdot (v + w) = u \cdot v + u \cdot w, (c u) \cdot v = c (u \cdot v) .

Geometric Interpretation

$u \cdot v > 0$ : acute angle; $= 0$ : right angle; $< 0$ : obtuse angle.
$∥ u + v ∥^{2} = ∥ u ∥^{2} + ∥ v ∥^{2} + 2 u \cdot v$ (law of cosines in vector form).
Projection extracts the component of $u$ parallel to $v$ .

Worked Example

Let $u = ⟨ 3, 4 ⟩$ and $v = ⟨ 1, 0 ⟩$ .

u \cdot v = 3, ∥ u ∥ = 5, ∥ v ∥ = 1,

\cos θ = \frac{3}{5}, {proj}_{v} u = \frac{3}{1} ⟨ 1, 0 ⟩ = ⟨ 3, 0 ⟩ .

Common Mistakes

Dividing by $∥ v ∥$ instead of $∥ v ∥^{2}$ in the projection formula.
Claiming $u \cdot v = ∥ u ∥ ∥ v ∥$ without $\cos θ$ (only if $θ = 0$ ).
Treating the zero vector as having a well-defined direction or unit vector.

Connections

Related: Lines and Planes, Distance and Midpoint
Next: Conic Sections; Orthogonality and Projections in linear algebra
Later: least squares via projections — Least Squares and QR

References

Vector algebra and the dot product follow OpenStax Calculus Volume 3.^[1]

OpenStax, Calculus Volume 3, https://openstax.org/details/books/calculus-volume-3 ↩︎