Vectors

The term vector is used in several different contexts, each coming with a slightly different definition. In basic linear algebra a vector often is considered a finite column of numbers. That’s the approach we follow here.

Definition

For \(d\in\mathbb{N}\) a \(d\)-tuple is an ordered list of real numbers, typically written as \((x_1,x_2,\ldots,x_d)\) with \(x_1,\ldots,x_d\) denoting the numbers. Here ‘ordered’ means that swapping two unequal numbers in the list yields a different \(d\)-tuple. Example: \((1,2,3)\neq(1,3,2)\). By \(\mathbb{R}^d\) we denote the set of \(d\)-tuples.

Vector is another term for \(d\)-tuple. In linear algebra vectors may be interpreted as points in space or as difference between two points (that is, describing a translation). Vectors often are written as columns:

\[\begin{equation*} x=\begin{bmatrix}x_1\\x_2\\\vdots\\x_d\end{bmatrix}. \end{equation*}\]

Length of a Vector

The (Euclidean) length of a vector \(x\) is defined as

\[\begin{equation*} |x|:=\sqrt{x_1^2+\cdots+x_d^2}. \end{equation*}\]

Sum of Vectors

Sums of vectors are defined componentwise:

\[\begin{equation*} x+y:=\begin{bmatrix}x_1+y_1\\\vdots\\x_d+y_d\end{bmatrix}. \end{equation*}\]

Multiples of Vectors

Products of real numbers and vectors are defined componentwise. For \(a\in\mathbb{R}\) and \(x\in\mathbb{R}^d\) we have

\[\begin{equation*} a\,x:=\begin{bmatrix}a\,x_1\\\vdots\\a\,x_d\end{bmatrix}. \end{equation*}\]

Inner Products

The inner product of two vectors \(x,y\in\mathbb{R}^d\) is

\[\begin{equation*} \langle x,y\rangle:=x_1\,y_1+\cdots+x_d\,y_d. \end{equation*}\]

Inner products are closely related to angles between vectors.

Outer Products

The outer product of two vectors \(x,y\in\mathbb{R}^3\) is

\[\begin{equation*} x\times y:=\begin{bmatrix}x_2\,y_3-x_3\,y_2\\x_3\,y_1-x_1\,y_3\\x_1\,y_2-x_2\,y_1\end{bmatrix}. \end{equation*}\]

The outer product yields a vector orthogonal to both factors.