A vector is an ordered list of numbers:

$$\mathbf{v} = \begin{pmatrix} 3 \ 1 \ 4 \end{pmatrix}$$

This is a vector in 3 dimensions (three entries). You can think of it as: - A point in space (coordinates 3, 1, 4) - An arrow from the origin to that point - A row of data (three measurements of something)

All three interpretations are useful. In statistics, you’ll mostly think of vectors as rows of data. In geometry, as arrows.

Operations

Vectors have two core operations:

Addition: Add corresponding entries.

$$(3, 1, 4) + (1, 5, 9) = (4, 6, 13)$$

Geometrically, this is “follow one arrow, then the other.” The result is the diagonal of the parallelogram formed by the two vectors.

Scalar multiplication: Multiply every entry by a number.

$$2 \times (3, 1, 4) = (6, 2, 8)$$

This stretches (or shrinks) the arrow without changing its direction. Multiply by $-1$ and the arrow flips to point the opposite way.

Explore a vector

Drag the purple dot to move the vector. The coordinates and length update in real time.

Vector addition

Drag the tips of u and v. The orange arrow is their sum $\mathbf{u} + \mathbf{v}$ — the diagonal of the parallelogram. Try making the vectors point in opposite directions and watch the sum shrink.

Length

The length (or norm) of a vector is its distance from the origin:

$$|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$$

For $(3, 4)$: $\sqrt{9 + 16} = \sqrt{25} = 5$. This is just Pythagoras extended to any number of dimensions.

A unit vector has length 1. You can turn any vector into a unit vector by dividing by its length: $\hat{\mathbf{v}} = \mathbf{v} / |\mathbf{v}|$. This keeps the direction but normalises the scale.

Dimensions

The number of entries in a vector is its dimension. A 2D vector lives in a plane. A 3D vector lives in space. A 67D vector — one entry per linguistic feature — lives in a 67-dimensional feature space. You can’t visualise it, but the arithmetic works the same way.