Vectors in n Dimensions

Lists of numbers

In the plane, a vector is an arrow given by two numbers. Nothing in the arithmetic of those numbers cares that there are two of them. A vector in ℝⁿ is an ordered list of $n$ real numbers, written as a column or, to save space, as a row:

$$\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} \qquad\text{or}\qquad \mathbf{v} = (v_1, v_2, \ldots, v_n)$$

The numbers $v_1, \ldots, v_n$ are the components. A price list for five products, the daily rainfall for a week, and the position of a point in space are all vectors. The entries mean different things; the arithmetic is the same.

Two operations define everything that follows. Both work one component at a time:

$$\mathbf{u} + \mathbf{v} = (u_1 + v_1, \ldots, u_n + v_n) \qquad c\mathbf{v} = (cv_1, \ldots, cv_n)$$

Adding needs both vectors to have the same number of components. The zero vector $\mathbf{0}$ has every component $0$, and $-\mathbf{v}$ means $(-1)\mathbf{v}$.

Why the rules carry over

Componentwise addition inherits its rules from the arithmetic of real numbers, one component at a time:

$$\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \qquad (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) \qquad c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$$

Every rule for vectors in ℝⁿ is a rule about real numbers applied in each component at once. So nothing breaks when $n$ grows past $3$ and the picture runs out.

Length and the dot product

The dot product extends the same way, by multiplying matching components and adding:

$$\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \cdots + u_nv_n$$

Taking $\mathbf{v} = \mathbf{u}$ gives the sum of the squares, so the length, or norm, is

$$\|\mathbf{u}\| = \sqrt{\mathbf{u} \cdot \mathbf{u}} = \sqrt{u_1^2 + \cdots + u_n^2}$$

In ℝ² this is the Pythagorean theorem, and in ℝⁿ it is the definition. Two vectors are orthogonal, the word for perpendicular in any dimension, when their dot product is $0$.

The standard basis

Let $\mathbf{e}_i$ be the vector with a $1$ in position $i$ and zeros elsewhere. In ℝ³,

$$\mathbf{e}_1 = (1, 0, 0) \qquad \mathbf{e}_2 = (0, 1, 0) \qquad \mathbf{e}_3 = (0, 0, 1)$$

Every vector splits into its components along these:

$$(3, -2, 5) = 3\mathbf{e}_1 - 2\mathbf{e}_2 + 5\mathbf{e}_3$$

Worked examples

Common mistakes

Practice problems