Vector
A vector is a mathematical object with both magnitude (size) and direction, represented by an arrow or as a list of components.
Formula
v = \langle v_1, v_2, v_3 \rangle
Definition
A vector has two properties: size (magnitude) and direction; an arrow pointing northeast with a certain length is a vector, while just a length with no direction is a scalar. Formally, a vector in $\mathbb{R}^n$ is an ordered $n$-tuple of real numbers $v = (v_1, v_2, \ldots, v_n)$, added component-wise, $u + v = (u_1+v_1, \ldots, u_n+v_n)$, and scaled, $cv = (cv_1, \ldots, cv_n)$, with magnitude $|v| = \sqrt{v_1^2 + \ldots + v_n^2}$. More abstractly, a vector space over field $F$ is a set $V$ with addition and scalar multiplication satisfying $8$ axioms (closure, associativity, commutativity, identity, inverses, distributivity); $\mathbb{R}^n$, polynomials, continuous functions, and matrices are all vector spaces, and linear maps between them are the morphisms of linear algebra.
Example
Wind blowing $20$ mph toward the north is a vector, while a temperature of $70$ degrees is a scalar (just a number, no direction). For $v = (3, 4)$: $|v| = \sqrt{9+16} = 5$; adding $u = (1, 2)$ and $v = (3, 4)$ gives $u + v = (4, 6)$, matching how displacement vectors add in physics. The space of continuous functions $C([0,1])$ is an infinite-dimensional vector space, and the Fourier series represents functions as infinite linear combinations of basis vectors $\sin(nx)$ and $\cos(nx)$.
Key Insight
Vectors let mathematics describe things that point somewhere: velocity, force, displacement, electric fields. Abstraction to vector spaces unifies diverse mathematics: PDE solutions form function spaces, quantum states are vectors in Hilbert space, and error-correcting codes are subspaces of $\mathbb{F}_2^n$.