The zero vector is considered to have zero magnitude and no specified direction in space. But if you take the limit of a nonzero vector when its magnitude approaches zero it had a defined direction in space. A lot of authors express that this will change the definitions you're working with but no author backs this up with an example.
My question is, how does vector algebra and so on change when you consider the zero vector as the limit of the magnitude of a nonzero vector?

 

Oh shoot, the title should read limit of a nonzero vector.
The best I've come up with is a null space (subspace) of a vector space. But the nonzero vector that become a zero vector in the limit had a specified direction.

Mod edit: fixed title for you.

 

The zero vector is considered to have zero magnitude and no specified direction in space. But if you take the limit of a nonzero vector when its magnitude approaches zero it had a defined direction in space. A lot of authors express that this will change the definitions you're working with but no author backs this up with an example.
My question is, how does vector algebra and so on change when you consider the zero vector as the limit of the magnitude of a nonzero vector?

I dont think vector algebra changes when you consider a zero vector as a limit to zero magnitude with a retained direction. Vector algebra requires a zero vector in order to define a vector space properly. The direction does nothing whether you consider it nonexistent, zero, a particular direction or all directions at once.

The only time when I've seen this idea of a zero vector with direction is useful is if one has a model of a point mass with an orientation vector. For example a car that is being steered might be modeled as a point mass, but the direction matters because the engine provides force in a particular direction. If one traces out a path of travel and a velocity vector is associated with the car and the path, then if the car stops, the velocity vector is zero with no direction. However, in this model we would wish to retain the direction that existed as the velocity approached zero. This is important information because if the car starts moving again, we need to know what direction and path it will start moving on, which depends on the last direction and curvature along the path.

In other words I believe the idea of a zero vector with direction can be applied in physics, but it really has no mathematical meaning.

 

Hi,

I think this is something you have to get from the application.

For example, in AC circuits we have a complex number which can be looked at as a vector:
a+bj => [a,b]

and the phase angle is:
atan2(b,a)

and because b and a might both be zero, we have to look at the circuit and try to figure out how a and b change and that gives us the angle. So we find the limit from the application not from the pure mathematics. This happens in other areas too. For example, if we have a real world problem that has a circle for solution, negative values would lead to unreal quantities so we would only use the upper half of the circle as solution. We cant get the information from the math itself though, we have to look at the application to determine what is applicable and what is not.