What type of vector is a force?

Question

According to my textbook there are two types of vectors: free vectors and position vectors. Free vectors can be moved without affecting the role they play in the problem, so they aren't attached to any particular location. On the other hand position vector or "tied" vector is a vector that is fixed relative to a given point, and is tied to that point.

This got me thinking: if these are really the only vectors then forces sometimes are not vectors, because if you imagine a beam and two parallel forces acting at either end, then:

The forces are not tied to any common point, so they can't be position vectors.

We aren't free to move them around since where they act on the beam is important - so they aren't free vectors either.

However, I was taught that forces are always vectors, so where is the problem?

Answer 1

Your book lies. But it's a very common lie that professors tell so that they can just package what they're trying to teach you into something you already know (or will know when you take Linear Algebra).

In reality, forces acting on particles are better described by translation vectors acting on an affine space. In an affine space, one distinguishes between points and translation vectors. These two objects don't "live" in the same space and so it really doesn't make sense to say that you can move translation vectors around. What you can do is apply them to any point you want. So, a better model of forces acting on a beam are translation vectors applied to points of an affine space.

I don't know your level of mathematical maturity, but this pdf doesn't look too difficult. Give it (or something similar) a read when you have the time.

Answer 2

On the other hand position vector or "tied" vector is a vector that is fixed relative to a given point, and is tied to that point.

This sentence makes no sense to me. But if you have a manifold $M$, there's a distinction between the points of $M$, which are basically locations in space, and the elements of $TM$, the tangent bundle, which can be thought of as the set of "based vectors."

Now we can make any vector space $V$ into a manifold $V$, and it's natural to identify the tangent bundle $TV$ with $V \times V$. We think of $(x,y) \in V \times V$ as a vector in the direction $y$ based at $x$. Under this viewpoint:

• Positions are points of the manifold, so they're elements of $V$.
• Velocities, momenta etc. are based vectors, so they're elements of $V \times V$.