'Directionality' of Vectors

Question

The correct mathematical definition of a vector $v$ is as an element of a vector space $V$ (with $V$ being the set of all $n$-tuples with respect to a field $F$)

Now the Physics interpretation of a vector is a 'mathematical object with both magnitude and direction'.

The biggest problem with the Physics intepretation of a vector is with 'directionality'. For example what direction does a vector have in $\mathbb{R^4}$?

But there is one particular case highlighting the interpretation of 'directionality' of a vector that is really bugging me. For example let's say I have $v \ni v \in \mathbb{R^1}$.

$$\implies v = \begin{bmatrix} \lambda \end{bmatrix} \ni (\lambda \in \mathbb{R})$$

In this example the sign of $\lambda$, which would be either $+$ or $-$, would be used in the Physics context to indicate whether the vector is 'pointing' left or right, along a line (a one-dimensional vector space).

But this brings about a contradiction, I believe, because if the sign of $\lambda$, where $\lambda$ is a scalar as $\lambda \in \mathbb{R}$, is used to determine direction, does that not imply that $-\lambda$ and $+\lambda$ are both vectors. With the sign denoting the 'direction' of the vector and $\lambda$ the 'magnitude'?

How do you reconcile the Mathematical definition of a vector with the Physics interpretation of it in this example then?

Or is there no contradiction and the Physics interpretation doesn't break down? This would be the case iff $\lambda$ is both a vector and scalar simultaneously, which would only occur iff the field which $\lambda \in \mathbb{F}$ (the field which $\lambda$ is in) forms a vector space over itself. I guess this must be the case as every field can form a vector space over itself.

Answer 1

If you consider $\lambda \in \mathbb{R}$, as a vector, the direction of $\lambda$ is $\lambda/\lvert \lambda\rvert$ which is either $+1$ or $-1$ (suppose $\lambda\neq 0$). The magnitude is $\lvert \lambda\rvert$ which is always positive. Notice that the direction of $\lambda$ could be $-1$.

Answer 2

The physics definition is not as you claim... that is only the Physics 101 definition. And just for the record, the mathematical definition of a vector is not as you claim. The best definition is that it is an element of a vector space, because then it must obey certain predefined axioms (I won't list them).

Now, vectors in $\mathbb{R^n}$ always have directionality with respect to some basis (yes, even when $n \geq 4$). If we are to consider $\mathbb{R^1}$ as a vector space over the field $\mathbb{R}$, then no such contradiction exists. Both $\lambda$ and $-\lambda$ are vectors, not scalars. Why? Because they live in the vector space $\mathbb{R^1}.$ In short, vectors are elements of vector spaces,and scalars are elements of whatever field the vector space lies over.

Last, the physics and mathematical definition of a vector and vector space are the same, and can be found under the "definition" link in https://en.wikipedia.org/wiki/Vector_space .

Answer 3

I think the ambiguity is arising here from a missing word.

A vector quantity is a physically measurable property having both magnitude and direction. This definition is used to distinguish between, for instance, distance and displacement, or velocity and speed. For brevity, it may often be abbreviated to just 'vector', but the object of interest in a physics context is much more specific than in broader mathematics.

Direction is relative. Most spaces of vector quantities afford an inner product such as the dot product, which extend the definitions of angles (and hence direction) to higher dimensions. Imagining directionality in $4D$ is not easy, but there are suitable analogies which allow you to imagine what this means.

@EmanuelePaolini's answer covers your misunderstanding about $1D$ vector quantities. This highlights that you must sometimes be careful to distinguish elements of a vector space and elements of the underlying field.