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I am trying to understand definition of length of a vector as directed line segment.

Sumarry:

So in building up definition we start from some space $E^3$ which is standard three dimensional space (we are using Hilbert axiomatization of Euclidian geometry).

Then we define oriented line segment as ordered pair of two points $(A,B)$ where $A,B \in E^3$ (we denote the oriented line segment as $\vec{AB})$.

Then we define relation of equivalence $\equiv$ (on $D=E^3 \times E^3$): two oriented line segmets $\vec{AB}$ nad $\vec{CD}$ are equivalent if the line segments $\overline{AD}$ and $\overline{BC}$ have common midpoint.

Now we call elements of quoitent space $V^3=D/_{\equiv} =\{[\vec{AB}] |\vec{AB} \in D \}$ vectors (denoted with small letters like $\vec{a}$).

Problem

How to define length of a vector constructed this way? My book says: length $|\vec{a}|$ of a vector $\vec{a}$ is understood as length of representant $\vec{AB}$ of vector $\vec{a}$, ie. number that we get from mesuring length of line segment $\overline{AB}$ with given unit length $\overline{OE}$. $$|\vec{a}| = \dfrac{\overline{AB}}{\overline{OE}} \in \mathbb{R}.$$

I do not understand what does this mean (what does mesuring mean and how to define length of line segment $\overline{AB}$ and what is length of line segmet $\overline{OE}$), so my question is: what does it mean? Can length of a vector by this definition be irrational, if it is only some (integer) number of basic unit lengths $\overline{OE}$ (if it really is just some number of basic unit lengths). Probably the main confusion is how does one define length of line segment (so that it can include all kind of real values using probably only Hilbert axioms because what else could be used here)?

Thank you for any help.

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  • $\begingroup$ See math.stackexchange.com/questions/2015726/… . Also I think your definition of relation $\equiv$ between oriented segments is not stardard i.e. your vectors won't be the same thing as what is usually called vectors $\endgroup$ – Kulisty Mar 1 '19 at 19:58
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A vector is a directed line segment. It has no magnitude, that is length.

Such a magnitude can be given in two ways, either through a covector or via a norm, and the latter can be given by an inner product and is what we generally think of, when we think of length.

We can however, say how much larger or smaller two parallel vectors are compared to each other. That is we have a notion of scale. This is not a magnitude, it has no absolute sense, but is a ratio.

(All this, by the way, generalises to tensors, they are rescalable directed area or volume elements).

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