How do we determine if a certain physical quantity is a vector?

For instance in Newtonian physics we treat position of objects, displacements, velocities, forces, momenta, angular velocities etc all as vector quantities (little arrows in space which have a certain direction & magnitude that can be added using the parallelogram law of vector addition & can be multiplied by scalars). But who said that these physical quantities can be modelled using vector algebra? Is this an empirical statement or is there a theoretical reasoning behind it?

For instance why do velocities add linearly in Newtonian Mechanics? If an object is given a velocity of 10m/s North & a velocity of 10m/s East, then according to what I've been taught, the 2 velocities will just add like vectors to give the net velocity of the object but I don't see how this statement is obvious in any way? Is there a deeper reason for why velocity ought to behave like a vector in Euclidean Space?

Moreover as I understand it, the vectors used in Newtonian Mechanics are vectors in 3 dimensional Euclidean Space whereas in relativity theory (where velocities don't add linearly) we use four vectors in an different abstract space, right? If relativity is the more fundamental theory, then why is it that these four vectors in abstract space behave like Euclidean vectors in real space for low velocities? Where's the connection?

When can we use vector algebra to model physical quantities? There are about 5 different definitions of vectors that I've encountered so far. How do we formally define vectors in physics?

Please be as elaborate as possible.

• Yes, all of this is based on empirical observations. One basically has to check that the axioms of vector spaces are approximated with very high precision by nature. How does one do that? By literally taking three yard sticks and by verifying Pythagoras. The Egyptians have done that for us, already, with a precision of better than 1%. With modern geodetic equipment we can do it to better than five, maybe six digits. The GPS system (by including relativistic corrections) does it probably to ten digits, or so (I could be wrong about that). Commented Apr 30, 2016 at 1:31
• Start with something you know is a vector, like a position ve ctor. Now take the derivative with respect to time: this automatically becomes a vector, thus velocity is a vector; repeat the process, and acceleration is a vector. Multiply mass by velocity to get momentum, so momentum is a vector. Take it's temporal derivativ to get the force, and it is a vector. But we already knew force is a vector because of the geometric addition rule, the parallogram rule. So it all ties together, math and physics of vectors. Commented Apr 30, 2016 at 1:34

A physical quantity is a vector if it transforms in the same way as a position vector when the coordinate system undergoes a transformation.

How do we formally define vectors in physics?

An excerpt from chapter one, page 12 of "Mathematics of Classical and Quantum Physics"

Originally, we introduced a vector as an ordered triple of numbers. The rule for expressing the components of a vector in one coordinate system in terms of its components in another system tells us that if we fix our attention on a physical vector and we rotate the coordinate system, the vector will have different numerical components in the rotate coordinate system.

So we are led to realize that a vector is really more than an ordered triple. Rather, it is many sets of ordered triples which are related in a definite way. One specifies a vector by giving three ordered numbers, but these numbers are distinguished from an arbitrary collection of three numbers by including the law of transformation under rotation of the coordinate frame as part of the definition. This law tells how all vectors change if the coordinate system changes.

Thus, one physical vector may be represented by infinitely many ordered triples. The particular triple depends on the orientation of the coordinate system of the observer. This is important because physical results must be the same regardless of one's vantage point, that is, regardless of the orientation of one's coordinate system.

This will be the case if a given physical law involves vectors on both sides of the equation. Now, from this point of view, the transformation rule and the orthogonality relations may be used to define vectors.

My intuitive answer is vectors are something with Vector additions and scalar multiplications.

Looking up my a textbook in algebra I think a more mathematician acceptable answer is 'vector space is a set, whose element can add together and multiplied by numbers. Vectors are the element of such set. The addition and multiplications must obey the following rules: (the exact phrasing is copied from Wikipedia: vector space)

1. Associativity of vector addition: u + (v + w) = (u + v) + w
2. Commutativity of vector addition: u + v = v + u
3. Identity element of vector addition: There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V.
4. Inverse elements of vector addition: For every v ∈ V, there exists an element −v ∈ V, called the additive inverse of v, such that v + (−v) = 0.
5. Compatibility of scalar multiplication with field multiplication: a(bv) = (ab)v
6. Identity element of scalar multiplication: 1v = v, where 1 denotes the multiplicative identity in F.
7. Distributivity of scalar multiplication with respect to vector addition: a(u + v) = au + av
8. Distributivity of scalar multiplication with respect to field addition: (a + b)v = av + bv '

So vectors are elements in vector space, you speak about Euclidean vectors, that is great! which means you observed that definitions of vectors changed as space changed. I don't know if that answers your question 'why do velocities add linearly in Newtonian Mechanics'. I think the reasoning is not 'velocities add linearly', its the other way around: we find that velocities add up, and can be multiplied by a number. And this 2 operation satisfied the 8 rules above, so we decided to use a vector space to describe those relations. As Physicists we simply borrows this mathematical tool to model something, like directions, force etc.

For more place refer to Wikipedia: vector space https://en.wikipedia.org/wiki/Vector_space or try a linear algebra textbook or open-course.

I don't know if my understanding is correct or of any help, still thanks for reading.

• In the context of classical physics, a "vector" is something more restrictive than the general mathematical definition of a vector space. In math terms, it's a vector space endowed with an action of the group $SO(3)$, which (effectively) gives a notion of "direction" in space to any vector. The "pure math" definition is used in quantum mechanics, but that's not what the OP was concerned with. Commented Feb 26, 2023 at 14:16