Zero Momentum in Elastic Collisions

[Vanadium 50]

The OP is having difficulty with the concept of a vector as it is first introduced, as in high school. Does anyone really think that the way to get him past this is to discuss abstract vector spaces? Really?

rkmurtyp, the zero vector is special. There is one of them, and since it's length is zero, the direction in which that length points is meaningless. If you are at rest, your velocity vector is zero. You are not moving north at the same time you are not moving east at the same time you are not moving south at the same time you are not moving west. You have one and only one velocity: zero.

 

[Dale]

The OP is having difficulty with the concept of a vector as it is first introduced, as in high school. Does anyone really think that the way to get him past this is to discuss abstract vector spaces? Really?

I do. I learned about the axioms of vector spaces in high school. The teacher spent a little time motivating vectors as arrows, but almost immediately afterwards showed us the axioms. It isn't that difficult and it resolves the problem the OP is having.

 

[D H]

I do. I learned about the axioms of vector spaces in high school.

Same for me. It's not that hard. All that is needed are those very simple axioms. They are not that hard.

V50, I intentionally avoided saying that the first few axioms make a vector space a commutative group. I was not trying to teach group theory. That is something to be learned much later in one's mathematical education.

This informal introduction to group theory (note well: without reference to the concept of "group theory") was, at least to me, very helpful in understanding the nature of vectors. That apparently is why my teacher back then decided to teach the axiomatic basis of vectors early on.

 

[Dale]

I think the approach was something like:
Arrows with length and direction
Axioms of vector spaces
Here is how you add two arrows
Here is how you multiply an arrow by a scalar
Show that those satisfy the axioms
Basis vectors ...

 

[megatyler30]

There is a zero vector, we talked about it in Calculus 3, the magnitude is zero (obviously) and equivalently you could say it is any direction and it's still the same vector.

 

[HallsofIvy]

There is a zero vector, we talked about it in Calculus 3, the magnitude is zero (obviously) and equivalently you could say it is any direction and it's still the same vector.

Yes, but the idea that the 0 vector can have "any direction" is precisely the thing that was confusing the original poster to begin with!

 

[megatyler30]

Well it's just the exception..

 

[BruceW]

It depends what is meant by the word "direction" and "parallel". The zero vector is not linearly independent of any other vector, so by this, I'd say the zero vector is parallel to any other vector, and has any direction. But it depends on how "parallel" and "direction" are defined. I don't know if there is any convention for this.