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Vectors



What Is a Vector
Notation and Representation
Types of Vectors
Vector Properties
Basic Vector Operations
Magnitude and the Norm
The Dot Product
The Cross Product
Linear Combinations
Summary: All Vector Operations at a Glance



Where Algebra Meets Geometry

A vector captures two pieces of information in a single mathematical object: how far and which way. This dual nature — part algebraic, part geometric — makes vectors the natural language for describing anything that requires both a size and an orientation. In R2\mathbb{R}^2 and R3\mathbb{R}^3, vectors are arrows that can be drawn, measured, and rotated. In Rn\mathbb{R}^n, they generalize beyond visual intuition into ordered lists of numbers that still obey the same algebraic rules. This section develops vectors from the ground up: what they are, how they behave, and the operations that give them structure.



What Is a Vector

At its core, a vector is a mathematical object defined by two attributes: a magnitude (a non-negative number representing size) and a direction (an orientation in space). A scalar, by contrast, is fully specified by a single number — temperature, mass, or area require no directional component. A vector demands both.

Geometrically, a vector appears as a directed line segment: an arrow drawn from an initial point to a terminal point. The arrow's length encodes the magnitude, and the way it points encodes the direction. Two arrows that share the same length and orientation represent the same vector, regardless of where in space they are positioned. This is the concept of a free vector — identity is determined entirely by magnitude and direction, never by location.

Algebraically, a vector in Rn\mathbb{R}^n is an ordered nn-tuple of real numbers:

v=(v1,v2,…,vn)\mathbf{v} = (v_1, v_2, \ldots, v_n)


Each entry viv_i is called a component. In R2\mathbb{R}^2, a vector has two components and can be plotted on a coordinate plane. In R3\mathbb{R}^3, three components place it in three-dimensional space. For n>3n > 3, direct visualization is no longer possible, but the algebraic framework continues to function identically. The components of a vector encode both its magnitude and its direction numerically, linking the geometric and algebraic viewpoints into a single coherent object.

Notation and Representation

Several notational conventions exist for vectors, and the choice often depends on context. Boldface lowercase letters are standard in printed text:

v,u,w\mathbf{v},\quad \mathbf{u},\quad \mathbf{w}


In handwritten work, an arrow placed above the letter serves the same purpose: v⃗\vec{v}, u⃗\vec{u}, w⃗\vec{w}. When a vector runs from a specific point AA to a specific point BB, the notation AB⃗\vec{AB} identifies both the direction and the endpoints explicitly.

The components of a vector can be written as a row or as a column. Row notation lists the entries horizontally:

v=(v1,v2,…,vn)\mathbf{v} = (v_1, v_2, \ldots, v_n)


Column notation stacks them vertically:

v=(v1v2⋮vn)\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix}


The distinction matters when vectors interact with matrices: matrix multiplication requires column vectors on the right side of the product. Throughout this section, both formats appear depending on which is clearer for the situation at hand.

Standard Basis Vectors


In any Rn\mathbb{R}^n, the standard basis consists of nn vectors, each with a 11 in exactly one position and 00 everywhere else. In R3\mathbb{R}^3, these are commonly written as i\mathbf{i}, j\mathbf{j}, and k\mathbf{k}:

i=(1,0,0),j=(0,1,0),k=(0,0,1)\mathbf{i} = (1, 0, 0), \quad \mathbf{j} = (0, 1, 0), \quad \mathbf{k} = (0, 0, 1)


In general Rn\mathbb{R}^n, the notation e1,e2,…,en\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n is used. Any vector can be decomposed as a sum of scaled basis vectors: v=v1e1+v2e2+⋯+vnen\mathbf{v} = v_1\mathbf{e}_1 + v_2\mathbf{e}_2 + \cdots + v_n\mathbf{e}_n. This decomposition makes the components explicit and connects directly to the idea of a linear combination.

Types of Vectors

Not all vectors play the same role. Several categories appear repeatedly throughout linear algebra, and recognizing them early simplifies everything that follows.

The Zero Vector


The zero vector 0\mathbf{0} has every component equal to zero. Its magnitude is 00, and it has no defined direction. Despite this, the zero vector is indispensable: it serves as the additive identity under vector addition, meaning v+0=v\mathbf{v} + \mathbf{0} = \mathbf{v} for every vector v\mathbf{v}.

Unit Vectors


A unit vector has magnitude exactly 11. It encodes pure direction with no scaling. The standard basis vectors are all unit vectors, but any nonzero vector can be converted into a unit vector through normalization — a process covered in detail on the magnitude page. Unit vectors are essential for isolating directional information from length.

Position Vectors


A position vector has its tail fixed at the origin and its head at a specific point in space. The vector p=(3,5)\mathbf{p} = (3, 5) as a position vector points from the origin to the point (3,5)(3, 5). This convention establishes a one-to-one correspondence between points and vectors: every point in Rn\mathbb{R}^n corresponds to exactly one position vector, and every position vector identifies exactly one point.

Free Vectors


A free vector is defined solely by its magnitude and direction, with no attachment to a particular location. Translating a free vector to a different starting point does not change the vector. Two arrows in different parts of the plane that share the same length and orientation represent the same free vector. Most of the algebra in this section treats vectors as free — their behavior under operations depends only on their components, not on where they happen to be drawn.

Equal Vectors


Two vectors are equal when all their corresponding components match: a=b\mathbf{a} = \mathbf{b} if and only if ai=bia_i = b_i for every ii. Geometrically, equal vectors have the same magnitude and the same direction. Position plays no part — a vector drawn at one corner of a diagram is identical to a vector drawn at another corner, provided the components agree.
Type Defining condition Magnitude Key feature
Zero vector 0 every component equals 0 0 additive identity (v + 0 = v); no defined direction
Unit vector ‖v‖ = 1 1 encodes pure direction; obtained from any nonzero vector by normalization
Position vector tail anchored at the origin, head at a specific point varies one-to-one correspondence between points in ℝⁿ and position vectors
Free vector identity determined by magnitude and direction alone varies location-independent — the same vector can be drawn anywhere in space

Vector Properties

Every vector carries intrinsic attributes — magnitude, direction, and dimensionality — that exist independently of any operation. Magnitude measures how large a vector is, direction specifies where it points, and dimensionality records how many components it contains. These are not consequences of addition or multiplication; they belong to the vector itself.

Beyond what individual vectors possess, vectors also relate to one another through structural relationships. Two vectors are equal when their components match exactly. They are parallel when one is a scalar multiple of the other, meaning they share the same or opposite direction. They are orthogonal when their dot product equals zero, indicating that they meet at a right angle in the geometric interpretation.

The properties page collects both the intrinsic and relational attributes in a single reference, with each property developed fully on the page where it naturally belongs.

Basic Vector Operations

Three operations form the algebraic backbone of everything in this section: addition, subtraction, and scalar multiplication. Each operates component by component, each produces a vector in the same Rn\mathbb{R}^n, and each has a geometric interpretation that reinforces the algebra.

Vector addition combines two vectors into a third by summing their corresponding components. Geometrically, this is the tip-to-tail construction — place the tail of the second vector at the head of the first, and the sum runs from the starting tail to the ending head. The parallelogram method provides an equivalent picture: the sum is the diagonal of the parallelogram formed by the two vectors.

Subtraction reverses the process: a−b\mathbf{a} - \mathbf{b} adds a\mathbf{a} to the negation of b\mathbf{b}, producing the vector that points from the tip of b\mathbf{b} to the tip of a\mathbf{a}.

Scalar multiplication scales a vector by a real number. Multiplying by a positive scalar stretches or compresses the vector without changing its direction; multiplying by a negative scalar also reverses it. These three operations satisfy a collection of algebraic rules — commutativity, associativity, distributivity — that make structured computation possible.

Magnitude and the Norm

The magnitude of a vector quantifies its length as a single non-negative number. In R2\mathbb{R}^2, this is the hypotenuse of the right triangle formed by the components. In R3\mathbb{R}^3, the same idea extends through a second application of the Pythagorean theorem. The general formula for any Rn\mathbb{R}^n is:

∥v∥=v12+v22+⋯+vn2\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}


This quantity is called the Euclidean norm, and it satisfies three fundamental properties: it is never negative, it scales predictably under scalar multiplication (∥cv∥=∣c∣∥v∥\|c\mathbf{v}\| = |c|\|\mathbf{v}\|), and it obeys the triangle inequality (∥a+b∥≤∥a∥+∥b∥\|\mathbf{a} + \mathbf{b}\| \leq \|\mathbf{a}\| + \|\mathbf{b}\|).

The norm also provides a natural notion of distance: the distance between two vectors a\mathbf{a} and b\mathbf{b} is ∥a−b∥\|\mathbf{a} - \mathbf{b}\|. Closely tied to magnitude is the concept of a unit vector — a vector whose norm equals 11. Any nonzero vector can be rescaled to unit length through normalization, a process that extracts pure direction by dividing out the magnitude.

The Dot Product

The dot product is the first operation in this section that does not return a vector. It takes two vectors and produces a scalar, computed by multiplying corresponding components and summing the results:

a⋅b=a1b1+a2b2+⋯+anbn\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \cdots + a_nb_n


This algebraic formula has a geometric equivalent: a⋅b=∥a∥∥b∥cos⁡θ\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\|\|\mathbf{b}\|\cos\theta, where θ\theta is the angle between the two vectors. The geometric form reveals what the dot product actually measures — how much two vectors align directionally. When the dot product is positive, the vectors point roughly the same way. When it is negative, they point roughly in opposite directions. When it equals zero, the vectors are orthogonal: perpendicular in the geometric sense.

This connection to angles makes the dot product the gateway to projection: decomposing one vector into a component along another and a component perpendicular to it.

The Cross Product

Where the dot product takes two vectors and returns a scalar, the cross product takes two vectors and returns a new vector. This returned vector is perpendicular to both inputs, giving the cross product a geometric role that the dot product cannot fill: it finds directions orthogonal to a given plane.

The cross product is defined exclusively in R3\mathbb{R}^3. For vectors a=(a1,a2,a3)\mathbf{a} = (a_1, a_2, a_3) and b=(b1,b2,b3)\mathbf{b} = (b_1, b_2, b_3):

a×b=(a2b3−a3b2, a3b1−a1b3, a1b2−a2b1)\mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2,\ a_3b_1 - a_1b_3,\ a_1b_2 - a_2b_1)


The magnitude ∥a×b∥=∥a∥∥b∥sin⁡θ\|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\|\|\mathbf{b}\|\sin\theta equals the area of the parallelogram that the two vectors span. The direction is determined by the right-hand rule, which introduces the concept of orientation — a handedness to three-dimensional space that has no analogue in the dot product. Unlike most operations encountered so far, the cross product is anti-commutative: swapping the order of the inputs reverses the direction of the result.

Linear Combinations

Every operation covered so far — addition, subtraction, scalar multiplication — is a special case of a single unifying idea: the linear combination. Given vectors v1,v2,…,vk\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k and scalars c1,c2,…,ckc_1, c_2, \ldots, c_k, the expression

c1v1+c2v2+⋯+ckvkc_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k


is a linear combination of those vectors. Adding two vectors is the special case where k=2k = 2 and both scalars equal 11. Scalar multiplication is the case k=1k = 1.

The set of all possible linear combinations of a given collection of vectors is called the span of that collection. The span of a single nonzero vector is a line through the origin. The span of two non-parallel vectors is a plane. The span of three non-coplanar vectors in R3\mathbb{R}^3 fills the entire space. Asking whether a particular vector lies in the span of a given set turns out to be equivalent to asking whether a certain system of equations has a solution — a connection that links vectors directly to the machinery of matrices and row reduction.

Linear combinations also set the stage for two concepts developed in the vector spaces section: linear independence, which identifies when no vector in a set is redundant, and basis, which captures the minimal spanning set needed to reach every point in a space.

Summary: All Vector Operations at a Glance

The operations introduced above share a unifying structure but vary widely in what they take in, what they return, and what they reveal about the underlying geometry. The table below collects every operation in this section, noting its inputs and output types, the formula, and its geometric meaning. Reading across each row situates a single operation in full; reading down each column compares operations on the same attribute — and each operation's name links to the dedicated page where it is developed in detail.
Operation Inputs Output Formula Geometric meaning
Vector addition two vectors in ℝⁿ vector in ℝⁿ v + w = (v₁ + w₁, …, vₙ + wₙ) tip-to-tail construction; diagonal of the parallelogram
Vector subtraction two vectors in ℝⁿ vector in ℝⁿ v − w = v + (−w) runs from the tip of w to the tip of v
Scalar multiplication scalar c and a vector vector in ℝⁿ cv = (cv₁, …, cvₙ) stretches or compresses; reverses direction when c < 0
Magnitude (norm) one vector in ℝⁿ non-negative scalar ‖v‖ = √(v₁² + … + vₙ²) length of the arrow; distance from origin
Dot product two vectors in ℝⁿ scalar v · w = Σ vᵢwᵢ = ‖v‖‖w‖ cos θ measures directional alignment; equals 0 iff vectors are orthogonal
Cross product two vectors in ℝ³ (only) vector in ℝ³ v × w = (v₂w₃ − v₃w₂, v₃w₁ − v₁w₃, v₁w₂ − v₂w₁) perpendicular to both inputs; magnitude equals parallelogram area
Linear combination k vectors and k scalars vector c₁v₁ + c₂v₂ + ⋯ + cₖvₖ unifying framework; the set of all such combinations is the span