Math Insight

Vectors in arbitrary dimensions

You may be familiar with two- and three-dimensional vectors (i.e., vectors in $\mathbf{R}^2$ and $\mathbf{R}^3$). For example, a two-dimensional vector $\vc{x}=(1,3)$ could describe the position of your feet in meters north and west of the corner of a room. A three-dimesional vector $\vc{x}=(1,3,1.5)$ could describe the location of your head, with the third dimension corresponding to distance from the floor.

We can generalize this concept to an arbitrary number of dimensions, say $n$ dimensions. We refer to an $n$-dimensional vector as a vector in $\mathbf{R}^n$ and write it as an n-tuple of numbers: \begin{gather*} \vc{x} = \mbox{$(x_1,x_2,x_3, \ldots, x_n)$}. \end{gather*} For example, $\vc{a}=(1,6,-23,0.23,0,400)$ is a vector in $\mathbf{R}^6$.

You might have trouble with high dimensional vectors if you are used to thinking of vectors as indicating a location in physical space, as in the above example for two- or thre-dimensional vectors. How can we talk about dimensions higher than three? A solution is to set aside the metaphor between the mathematical dimensions of a vector and the spatial dimensions of the world we live in. This metaphor is great to help us visualize the number line in one-dimension, or the space of three-dimensional vectors. But for many of us, this metaphor may hamper our efforts to delve into higher dimensions, as we are restrained by our experience of just three spatial dimensions in the world around us. It may be easier to think of a high dimensional vector as simply describing quantities of distinct objects. For example, a five-dimensional vector could describe the numbers of apples, oranges, banana, pears, and cherries on the table.

High-dimensional vectors have a lot of practical use. There are not just an abstraction that mathematicians invented so they could discuss ideas others can't understand. You can read examples of why we often need to study $\mathbf{R}^n$ for $n$ larger than 3.