What is abstraction of direction in considering vectors such as used in Engineering & Physics?

Question

In the use of vectors of engineering and physics, we encounter objects that obey the axioms of a vector space but also have two new attributes of length (or, magnitude) and direction (e.g. direction in space).

In the study of vector spaces, you do not encounter these notions of length and direction. To obtain the idea of length, you need to consider a vector space with added structure to include the concept of normed vector elements as commonly displayed in Banach Space or Hilbert space (or others). Indeed, other attributes such as a metric (e.g. Banach), and inner product can be included.

But, where does the notion of direction come in? The closest I get is the idea of using direction cosines derived from the inner product of an N-dimensional space of vectors. But, is this the only abstraction of direction (which is not much of an abstraction to me).

I am not sure what I am looking for but I guess maybe if someone could name a formal algebraic structure (extension of Banach Space?) where direction is included as one of the added structures and then I could use that to study.

Answer 1

Well we already seem to have a structure that encompasses length and direction in the notions of $\mathbb{R}^2$ and $\mathbb{R}^n$.

Describe a vector in $\mathbb{R}^2$ by $(r,\theta)$ where $r$ is the length of the vector and $\theta$ is the direction of the vector.

In $\mathbb{R}^n$ we can do the exact same thing, just using the $n$-tuple

$$(r,\theta_1,\ldots,\theta_{n-1})$$ so that the $\theta_i$'s give all necessary information about direction.

Indeed it is not necessary for a general 'weird' vector space to have a well-defined notion of direction, as is the case of $C^{\infty}_c(X)$, all infinitely differentiable functions of compact support over a topological space $X$.

The tangent space of a smooth manifold $M$, denoted $TM$ is an example of an affine space in which there is a notion of 'direction', relative to each point $x \in M$.

Perhaps define direction as being invariant under simple translations? For example, if I move a vector in $\mathbb{R}^n$ without rotation, surely I preserve direction. Maybe even go as far as to say that $e^x$ and $e^x+c$ surely have the 'same direction' in $C^{\infty}(X)$.

Therefore it may make sense to define length as that which is invariant under rotation and translation but not scaling, while direction is that which is invariant under translation and scaling but not rotation.

Answer 2

In my opinion direction can be defined only fo Hilbert spaces ( sice these are the spaces where we cand define an angle).

If $V$ is an Hilbert space than, given two vectors $v,w$ we can say that they have the same direction ad are equi-oriented iff $ \langle v,w\rangle=||v||\cdot||w||$ and have the same direction but are opposite iff $ \langle v,w\rangle=-||v||\cdot||w||$.

This seems a good definition that works also for infinite dimensional Hilbert spaces.

Answer 3

Relative directions of elements of vector spaces can be expressed via the scalar product: If we have a (pre-)Hilbert space $\cal H$ with a scalar product $\langle \cdot, \cdot \rangle$, then we say that two elements $f,g \in \cal H$ are orthogonal (i.e. their relative angle is 90°) if $\langle f, g \rangle = 0$.

If you take a (say, finite) set of vectors in $\cal H$, you can find a set of vectors which have the same span $\cal M$ as your original set, but its elements are piecwise orthogonal. You can do this with the Gram-Schmidt process, ending up with an orthogonal basis of $\cal M$. Thus, a scalar product helps you to 'establish' something like a coordinate system.

Furthermore, if you are not interested in finite-dimensional subsets, but the whole space, then you can start to look at complete orthonormal systems (sets of vectors which have norm equal to 1, are pairwise orthonormal and whose span is the whole Hilbert space).

Now to get back to the original geometric interpretation. If you have a scalar product and a complete orthonormal system $\{e_1, e_2, \dots\}$ of a Hilbert space (i.e. a coordinate system), then you can take a vector $f\in H$ and define

$$\arccos \left(\frac{\langle f,e_i\rangle}{\langle f,f \rangle^{1/2}}\right)$$

to be the angle between the 'direction' of $e_i$ and $f$. If the angle is 0°, then you can look at the term inside the $\arccos$ to see if they point in the same direction (it has the value +1) or in the opposite direction (it has thevalue -1).