A basic question about vectors

[mantrapad]

I've found this regular definition of vectors: A quantity which has both magnitude and direction.

On some sites, I've also found the addition: They satisfy the vector laws of addition (commutative, associative, distributive).

Is it really necessary that all vectors should satisfy all those laws? Are there any exceptions?

 

[James Leighe]

I've found this regular definition of vectors: A quantity which has both magnitude and direction.

On some sites, I've also found the addition: They satisfy the vector laws of addition (commutative, associative, distributive).

Is it really necessary that all vectors should satisfy all those laws? Are there any exceptions?

Yes and no respectively. Otherwise it wouldn't be a vector!
There are other things that are similar to vectors, but we give them different names (like scalars and stuff).

 

[Tac-Tics]

I've found this regular definition of vectors: A quantity which has both magnitude and direction.

On some sites, I've also found the addition: They satisfy the vector laws of addition (commutative, associative, distributive).

Is it really necessary that all vectors should satisfy all those laws? Are there any exceptions?

The "magnitude and direction" definition is an intuitive one. They make a lot of sense for R^n, but for some vector spaces, it doesn't. There are function vector spaces, where we create rules for adding and scaling functions. For example, if f(x) = x^2 and g(x) = x + 1, then (f + g)(x) = x^2 + x + 1. But what "direction" does f, g, or f+g point? It's much more abstract!

 

[mantrapad]

Thanks for your replies and sorry about my late reply.

I was also wondering - current is neither a scalar nor a vector, right? But we say "current flows from A to B", which specifies a direction right? So it should be a vector...

And also, if it is neither, then which category does it fall in?