Zero Momentum in Elastic Collisions

[D H]

rkmurtyp, I strongly recommend you read the rules of this site. When you argue against the uniqueness of the zero vector you are, perhaps unknowingly, violating the rules of this site. If you are doing this knowingly you will not last long here. Our fundamental goal at this site is to help students learn. That's a bit tough when the student is being obstinate. This obstinacy is your main problem. It is keeping you from learning.

Here are the rules for what it means for something to be a vector. An element of a set V is a vector v (and the set V is a vector space) if the set is endowed two operations, addition of elements of V and scalar multiplication by an element of a field F, and if the set has an additive identity 0, subject to the following:

1. Vector addition must be:
   • Closed: u+v is an element of V for all u, v in V,
   • Commutative: u+v = v+u for all u, v in V, and
   • Associative: (u+v)+w=u+(v+w) for all u, v, w in V.
2. 0 is an additive identity: v+0=v for all v in V.
3. Every vector v in V has an additive inverse: v+(-v)=0.
4. Scalar multiplication must be:
   • Closed: av is an element of V for all a in F and all v in V,
   • Associative: a(bv)=(ab)v for all a, b in F and all v in V,
   • Distributive: a(u+v)=au+av and (a+b)v=av+bv for all a, b in F and all u, v in V.
5. The field's multiplicative identity is a multiplicative identity for scalar multiplication: 1v=v.

Note well: The concept of magnitude (or length) and direction is not in the above list. Nor is the concept of an inner product (or dot product). The existence of these concepts are not necessary conditions for something to qualify as a "vector". On the other hand, the existence of a (unique) zero vector is essential.

The zero vector is provably unique. Assume there exists some vector 0'≠0 that also satisfies condition #2. Now consider the sums 0+0' and 0'+0. By condition #2, 0+0'=0 and 0'+0=0'. Since addition is commutative, 0+0'=0'+0. This means 0'=0, which violates the assumption that 0'≠0. This is a contradiction, and hence the assumption that there exists some some vector 0'≠0 that also satisfies condition #2 is false. In other words, 0 is unique.

The naive definition of a vector as something with length and direction is a specialization of the general concept of a vector. When you look at vectors from this narrower perspective, the zero vector is still there, and it is still unique. For vector spaces that have this concept of length and direction, the zero vector has a well defined length (0), but its direction is necessarily undefined.

 

[voko]

Here are the rules for what it means for something to be a vector

With all due respect, this is not the only definition possible.

Vector

geometric

A directed segment of a straight line in a Euclidean space, one end of which (the point A) is said to be the origin, while the other (the point B) is said to be the end of the vector.

http://www.encyclopediaofmath.org/index.php/Vector

A similar definition is given in Ito's Encyclopedic Dictionary Of Mathematics:

http://books.google.ru/books?id=azS2ktxrz3EC&pg=PA1678#v=onepage&q&f=false

The very first treatise on vectors, Gibbs's Elements of Vector Analysis, opens with the following passage:

1. Definition.—If anything has magnitude and direction, its magnitude and direction taken together constitute what is called a vector.

The numerical description of a vector requires three numbers, but nothing prevents us from using a single letter for its symbolical designation. An algebra or analytical method in which a single letter or other expression is used to specify a vector may be
called a vector algebra or vector analysis.

Def.—As distinguished from vectors the real (positive or negative) quantities of ordinary algebra are called scalars.

As it is convenient that the form of the letter should indicate whether a vector or a scalar is denoted, we shall use the small Greek letters to denote vectors, and the small English letters to denote scalars. (The three letters, i, j, k, will make an exceptin, to be mentioned more particularly hereafter. Moreover, $\pi$ will be used in its usual scalar sense, to denote the ratio of the circumference of a circle to its diameter.)

2. Def.—Vectors are said to be equal when they are the same both in direction and in magnitude. This equality is denoted by the ordinary sign, as $\alpha = \beta$. The reader will observe that this vector equation is the equivalent of three scalar equations.

A vector is said to be equal to zero, when its magnitude is zero. Such vectors may be set equal to one another, irrespectively of any considerations relating to direction.

3. Perhaps the most simple example of a vector is afforded by a directed straight line, as the line drawn from A to B. We may use the notation AB to denote this line as a vector, i.e., to denote its length and direction without regard to its position in other
respects. The points A and B may be distinguished as the origin and the terminus of the vector. Since any magnitude may be represented by a length, any vector may be represented by a directed line; end it will often be convenient to use language relating to vectors, which refers to them as thus represented.

You termed such definitions "naive", but, frankly, I see nothing naive in them, especially in the context of 3D physics. Would you care to explain why they are "naive"?

 

[rkmurtyp]

rkmurtyp, I strongly recommend you read the rules of this site. When you argue against the uniqueness of the zero vector you are, perhaps unknowingly, violating the rules of this site. If you are doing this knowingly you will not last long here. Our fundamental goal at this site is to help students learn. That's a bit tough when the student is being obstinate. This obstinacy is your main problem. It is keeping you from learning.

Here are the rules for what it means for something to be a vector. An element of a set V is a vector v (and the set V is a vector space) if the set is endowed two operations, addition of elements of V and scalar multiplication by an element of a field F, and if the set has an additive identity 0, subject to the following:

1. Vector addition must be:
   • Closed: u+v is an element of V for all u, v in V,
   • Commutative: u+v = v+u for all u, v in V, and
   • Associative: (u+v)+w=u+(v+w) for all u, v, w in V.
2. 0 is an additive identity: v+0=v for all v in V.
3. Every vector v in V has an additive inverse: v+(-v)=0.
4. Scalar multiplication must be:
   • Closed: av is an element of V for all a in F and all v in V,
   • Associative: a(bv)=(ab)v for all a, b in F and all v in V,
   • Distributive: a(u+v)=au+av and (a+b)v=av+bv for all a, b in F and all u, v in V.
5. The field's multiplicative identity is a multiplicative identity for scalar multiplication: 1v=v.

Note well: The concept of magnitude (or length) and direction is not in the above list. Nor is the concept of an inner product (or dot product). The existence of these concepts are not necessary conditions for something to qualify as a "vector". On the other hand, the existence of a (unique) zero vector is essential.

The zero vector is provably unique. Assume there exists some vector 0'≠0 that also satisfies condition #2. Now consider the sums 0+0' and 0'+0. By condition #2, 0+0'=0 and 0'+0=0'. Since addition is commutative, 0+0'=0'+0. This means 0'=0, which violates the assumption that 0'≠0. This is a contradiction, and hence the assumption that there exists some some vector 0'≠0 that also satisfies condition #2 is false. In other words, 0 is unique.

The naive definition of a vector as something with length and direction is a specialization of the general concept of a vector. When you look at vectors from this narrower perspective, the zero vector is still there, and it is still unique. For vector spaces that have this concept of length and direction, the zero vector has a well defined length (0), but its direction is necessarily undefined.

Thanks
I assure you, I have no intention of flouting any rules of the forum. It is such a useful forum to learn, to share and discuss what we know and so on. One should be happy to be a part of the forum.

I don't want to be obstinate either. My urge to understand things in an elementary fashion, perhaps, gave you that impression.

It is better, I think, I don't pursue this thread further.

 

[atyy]

Yes, it looks as if a special rule is required to define a zero vector.

Yes, in D H's post #31, this special rule is his second rule:
"0 is an additive identity: v+0=v for all v in V."

Since vectors are dimensional quantities (having units), when we superpose two vector diagrams (which I think is not forbidden) say, of forces and the accelerations or velocity changes they produce, the zero vectors create problems because of the so called uniquness of a zero vector.

The zero vector is not unique in that sense. While you can add and subtract two forces, you cannot add and subtract a force and a velocity. The vectors for force and velocity live in different vector spaces, with different zero vectors.

 

[BruceW]

Please tell me: When are two vectors said to be linearly independent?

For two vectors in Euclidean space, they are linearly independent when they do not lie on the same line. For example, if you have two linearly independent vectors v and w then the linear combination av+bw can be used (with scalars a and b) to give any vector in 2D Euclidean space. If v and w were not linearly independent, then you could only make some subset of 2D Euclidean space, using them.

Therefore, the zero vector 0 is not linearly independent to any other vector. Using some vector v and the zero vector, the possible combinations you can make using av+b0 are no different to the possible combinations you could make using only the vector v, without using the zero vector (i.e. av).

 

[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.