Difference between Euclidean space and vector space?

Question

I often hear them used interchangeably ... they are very complicated to make any use of.

Wikipedia words:

Euclidean space:

One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle.

Vector space:

A vector space is a mathematical structure formed by a collection of elements called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context.

They are not related at all. A vector space is a structure composed of vectors and has no magnitude or dimension, whereas Euclidean space can be of any dimension and is based on coordinates.

I hear 3-D programming uses vectors, so Euclidean geometry should be useless, no?

Basically, aren't they unrelated?

Answer 1

A concise mathematical term to describe the relationship between the Euclidean space $X = \mathbb E^n$ and the real vector space $V = \mathbb R^n$ is to say that $X$ is a principal homogeneous space (or ''torsor'') for $V$. This is a way of saying that they are definitely not the same objects, but they very much are related to each other.

In particular:

• The objects $\mathbb E^n$ and $\mathbb R^n$ are exactly the same as sets of elements -- they both correspond bijectively to $n$-tuples of real numbers.

However,

• in the vector space $\mathbb R^n$ we are allowed to add any two vectors (using the ''tip to tail'' visualization), whereas in Euclidean space $\mathbb E^n$ there is no natural way to describe the process of ''adding'' two points. Instead, given two points $P,Q$ in $\mathbb E^n$ we can naturally define their difference $\vec{v} = P-Q$, which is a vector in $ \mathbb R^n$. This vector tells us how to get from point $Q$ to $P$ in $\mathbb E^n$.

• in $\mathbb R^n$ there is a special ''zero vector'' $\vec{0} = (0,\ldots,0)$ which satisfies the additive property $\vec{0} + \vec{v} = \vec{v}$ for any $\vec{v}\in \mathbb R^n$, while in $\mathbb E^n$ there is no point that is somehow more special that the other ones -- i.e. the space is ''homogeneous'' meaning it looks the same around every point.

• for a vector $\vec{v}\in \mathbb R^n$ we can compute its length (or ''magnitute'' or ''norm'' etc.) by the formula $$|\vec{v}| = \sqrt{v_1^2 + \cdots + v_n^2}.$$ For a point $P\in \mathbb E^n$, it does not make sense to ask what its length or distance is. It only makes sense to ask the distance between two points $P,Q$.

There are many natural examples of torsors motivated by physics, discussed in this blog post of John Baez, as well as more rigorous definitions if you are interested.

Answer 2

While a vector space is something very formal and axiomatic, Euclidean space has not a unified meaning. Usually, it refers to something where you have points, lines, can measure angles and distances and the Euclidean axioms are satisfied. Sometimes it is identified with $\mathbb{R}^2$ resp. $\mathbb{R}^n$ but more as an affine and metric space (you have both points and vectors, not just vectors). So, the Euclidean space has softer meaning and usually refers to a richer structure.

Answer 3

The vectors in vector spaces are abstract entities that satisfy some axioms.

For a vector space that is not a set of points, consider the set of all continuous function $[0,1] \to \mathbb R$.

The Euclidean spaces $\mathbb R^n$ are examples of vector spaces.

Answer 4

An Euclidean space $\mathbb E^n$ can be defined as an affine space, whose points are the same as $\mathbb R^n$, yet is acted upon by the vector space $(\mathbb R^n, +, \cdot)$. If you select a point $a \in \mathbb E^n$, you can define a vector space $\mathbb E^n_a$ which has $a$ as the origin, by mapping $b \mapsto b - a$. Then an inner product can be defined as usual. Alternatively, if you only need distances, you can define a metric $d$ on $\mathbb E^n$ by $$d(a, b) = \|a - b\|,$$ where $\|\cdot\|$ is the Euclidean norm.

Answer 5

Vector space is just 'space' of objects following certain rules: a+b=b+a, (a+b)+c=a+(b+c), k(a+b) = ka + kb, and so on. k should be from another 'space' with certain rules, such as ℝ

Normed vector space is a vector space with "norm" function defined, ||a||, which corresponds to our intuitive "size of a vector" and follows simple rules: ||a+b||<=||a||+||b||, etc.

Euclidean space is the Normed Vector Space with coordinates and with Euclidean Norm defined as square root of sum of squares of coordinates; it corresponds to our intuitive 3D space, Pythagorean theorem, angle between lines definition, 5th axiom of Euclid (and Minkovski provided example of different space where 5th axiom is not true), and so on; it corresponds to geometry which we learn at school and touch in real life.

There are many other examples of normed vector spaces: Minkovski (used in Einstein theory of relativity), Banach, Hilbert, etc.

Answer 6

There's not a unique definition for Euclidean space, but usually it's the set of all n-tuples of real numbers $R^n = { {(x_1, ..., x_n) : x_i \in R} } $ with some 'structure' define on it that allows one to measure angles and lenghts, like a inner product space or a metric. The set $R^n $ with it's usual operations of sum and scalar multiplication is a example of vector space over $R $, and it can be shown that every finite-dimensional vector space over $R $ is isomorphic to some Euclidean Space, meaning that in pratice every vector space (of finite dimension) is 'equal' to a Euclidean Space