Vector

A vector $\mathbf{x}=(x_1,x_2,\dots ,x_n)$ is an ordered collection of $n$ real numbers, conventionally written as an $n\times 1$ column matrix, that represents a point or direction in $n$-dimensional space ${\mathbb{R}}^n$. In statistics, vectors represent a single multivariate observation (one row of a dataset, one value per variable), a full sample of $n$ observations on one variable, a set of model coefficients, or a random draw from a multivariate distribution, and the vector-space operations of addition and scalar multiplication carry over directly, allowing entire samples or parameter sets to be manipulated as single algebraic objects rather than component by component.

• A vector can be read two ways depending on context: as a single multivariate observation (one row of a data table, with one entry per variable) or as an entire sample on a single variable (one entry per observational unit) -- both uses are standard, and which one applies is determined entirely by context, not by the vector's algebra.
• Two vectors of the same dimension add componentwise, and a vector scaled by a number (scalar multiplication) stretches or shrinks every component by that same factor -- together these two operations define a vector space, the structural setting every other linear-algebra concept in this category builds on.
• The dot product of two vectors, $\mathbf{x}\cdot \mathbf{y}={\sum }_i{x}_i{y}_i$, underlies covariance and correlation computations, projections, and the definition of orthogonality (two vectors are orthogonal exactly when their dot product is zero) -- one of the most frequently reused building blocks in multivariate statistics.