Length and distance

Vectors: Distance, angle, dot product and cross product

Length and distance

Length in 2D and 3D

We denote the length of a vector $\vec{v}=\cv{v_1\\ v_2}$ in the coordinate plane $\mathbb{R}^2$ , more precisely the Euclidean length, by $\lVert\vec{v}\rVert$ . It can be calculated via the Pythagorean theorem: $\lVert\vec{v}\rVert=\sqrt{v_1^2+v_2^2}$

When we deal with a vector $\vec{v}=\cv{v_1\\ v_2\\ v_3}$ in $\mathbb{R}^3$ , the notion of length becomes $\lVert\vec{v}\rVert=\sqrt{v_1^2+v_2^2+v_3^3}$

Examples

$\begin{aligned}\left\lVert\cv{2\\ -3}\right\rVert &=\sqrt{2^2+(-3)^2} \\ &= \sqrt{4+9}=\sqrt{13} \\ \\ \left\lVert\cv{2\\ 3\\ -z}\right\rVert &=\sqrt{2^2+3^2+(-z)^2} \\ &= \sqrt{4+9+z^2}=\sqrt{13+z^2}\end{aligned}$

This can be generalized to the $n$ -dimensional coordinate space $\mathbb{R}^n$ .

The Euclidean length of a vector $\vec{v}=\cv{v_1\\ \vdots \\ v_n}$ in the $n$ -dimensional space $\mathbb{R}^n$ , also called the (Euclidean) norm, we denote by $\lVert\vec{v}\rVert$ . It can be calculated as follows: $\lVert\vec{v}\rVert=\sqrt{v_1^2+\cdots + v_n^2}=\sqrt{\sum_{i=1}^n v_i^2}$

We write Euclidean length and norm, because in the coordinate space, other lengths can be defined. For example, you can get define in $\mathbb{R}^n$ also the 1-norm as $||\vec{v}||=|v_1|+\cdots |v_n|\text{.}$ Here, the absolute value of a real number $a$ is denoted by $|a|$ . The 1-norm has similar properties as the Euclidean norm.

More generally, we can define for any natural number $p$ the $p$ -norm of a vector $\vec{v}=\cv{v_1\\ \vdots \\ v_n}$ in $\mathbb{R}^n$ as $\lVert|\vec{v}\rVert|=\sqrt[p]{|v_1|^p+\cdots+ |v_n|^p}=\sqrt[p]{\sum_{i=1}^n |v_i|^p}$ Thus, the Euclidean norm is in fact the $2$ -norm.

The distance between two vectors $\vec{u}$ and $\vec{v}$ is noted as $d(\vec{u}, \vec{v})$ . This is by definition the length of the difference vector $\vec{u}-\vec{v}$ , so $d(\vec{u}, \vec{v})=\lVert\vec{u}-\vec{v}\rVert$ .

The distance between two points $P$ and $Q$ in the plane or in space is equal to the length of the vector $\vec{PQ}$ and corresponds to the distance between the vectors $\vec{OP}$ and $\vec{OQ}$ , where $O$ is the origin.

In coordinates, the Euclidean distance between the vectors $\vec{u}=\cv{u_1\\ \vdots\\ u_n}$ and $\vec{v}=\cv{v_1\\ \vdots\\ v_n}$ can be calculated as $d(\vec{u}, \vec{v})=\sqrt{(u_1-v_1)^2+(u_2-v_2)^2+\cdots+(u_n-v_n)^2}\tiny.$

A vector $\vec{e}$ is called a unit vector if its norm (i.e., its length) is equal to $1$ : $\lVert\vec{e}\rVert=1$ . Division of a nonzero vector $\vec{v}$ by its norm, that is, multiplication by the reciprocal value of the norm, leads to a unit vector $\vec{e}=\frac{1}{\|\vec{v}\|}\cdot \vec{v}$ with the same direction as $\vec{v}$ , but with length equal to $1$ .

the Euclidean norm $\mathbb{R}^n$ has the following properties: $\begin{aligned} \lVert\vec{v}\rVert > 0 & \quad\text{if } \vec{v}\neq\vec{0}\\ \lVert\vec{v}\rVert = 0 & \quad\text{if } \vec{v}=\vec{0}\\ \lVert c\cdot\vec{v}\rVert =|c|\cdot \lVert \vec{v}\rVert & \quad\text{voor alle }c \in \mathbb{R}\\ \lVert\vec{u}+\vec{v}\rVert \leq \lVert\vec{u}\rVert+\lVert\vec{v}\rVert & \quad\text{for all } \vec{u}, \vec{v}\in \mathbb{R}^n\end{aligned}\text{.}$

The properties are exactly what is needed to turn a vector space $V$ via norm $\lVert\;\rVert : V\longrightarrow \mathbb{R}\;$ into a so-called normed vector space.

The last inequality is called the triangle inequality or Minkowski inequality.

Two vectors $\vec{u}$ and $\vec{v}$ in $\mathbb{R}^2$ or $\mathbb{R}^3$ are perpendicular to one another if and only if $\lVert\vec{u}\rVert^2 + \lVert\vec{v}\rVert^2= \lVert\vec{u}-\vec{v}\rVert^2\tiny.$

This follows directly from the application of the Pythagorean theorem to the triangle with the origin and the two terminal points of the vectors as vertices.

Let $U$ be a line or a plane in space and $P$ a point. There is a unique point $Q$ on $U$ that has the shortest distance from all points on $U$ to $P$ . This point is characterized by the property that the vector $\vec{PQ}$ is perpendicular to the direction vector(s) of $U$ state. The point $Q$ is called the perpendicular projection or orthogonal projection of $P$ on $U$ .

In the visualisation below, $U$ is the span of the vector $\vec{u}$ and we have drawn a perpendicular projection of a vector $\vec{v}$ on the line $U$ . The vector $\vec{w}$ is a scalar multiple of the vector $\vec{u}$ . The length of the vector $\vec{v}-\vec{w}$ is minimal when this vector is perpendicular to the line $U$ . By dragging the terminal point of the vector $\vec{w}$ along the line you can explore this.

GeoGebra

In the visualisation below, $U$ is a plane through the origin and we have drawn a perpendicular projection of a vector $\vec{P}$ on the plane $U$ . The length of the vector $\vec{QP}$ is minimal under the lengths of $\vec{RP}$ for all choices of the vectors $\vec{R}$ in the plane.

GeoGebra

Calculate the exact Euclidean length of the vector $\vec{v} = \cv{4 \\ -4 \\ -1}$ .

$\lVert\vec{v}\rVert = {}$ $\sqrt{33}$

The length of a vector $\cv{x \\ y \\ z}$ is given by $\sqrt{x^2+y^2+z^2}$ .
In the case of $\vec{v} = \cv{4 \\ -4 \\ -1}$ , and thus:
$\begin{aligned} \|\vec{v}\| &= \sqrt{4^2 + (-4)^2 + (-1)^2} \\ \\ &= \sqrt{16 + 16 + 1} \\ \\ &=\sqrt{33} \end{aligned}$

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