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Zero Vector

  1. Dec 30, 2013 #1
    Is it possible to have a zero vector? The question arises in the context of conservation of momentum. In center of mass reference frame, the momentum of two masses involved in an elastic collision is zero before (as well as after) collision. It is because the two masses have equal and oppposite values of momenta. Together they have zero momentum. What is the direction of this zero momentum? Since there are two possible directions opposite to one another which one do we choose. If we choose one of those directions what valid reason can we have to reject the opposite direction?
  2. jcsd
  3. Dec 30, 2013 #2


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    Yes it is possible to have a zero vector, and it has no direction - for roughly the same reason that zero is neither a positive number nor a negative number.
  4. Dec 30, 2013 #3


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    In computer science it is both.
  5. Dec 30, 2013 #4
    Not only is it possible, it is necessary by definition that any vector space V (that is, even if you don't consider eg vectors in 3d-space, but more general vectors) has a neutral element of sum, denoted (here) by 0 and defined s.t. v+0=v for all v∈V, that is, a zero vector.

    In fact, a vector space (such as R3) can be defined by considering something called an (Abelian) group equipped with additional structure, and it is required that every group G has an identity element e s.t. e.g=g.e=g, for all g∈G where . is something called the law of composition (it's the addition of vectors in the case of a vector space). If the group is equipped with the additional structure to make it a vector space, the e here is the zero vector of the vector space. (As a consequence of the definition of the identity, it turns out that it (and therefore the zero vector) is also unique, that is, there always exists exactly one zero vector in a specific vector space.)
  6. Dec 30, 2013 #5
    Please tell me: What is the difference between zero vector and zero scalar? If zero vector has no specific direction it is no different from a scalar zero and then we cannot add that scalar to any vecotor because addition of a vector and a scalar is forbidden. On the other, hand if zero vector has a specific direction how do we know which direction it is.

    If possible an explanation with reference to binary elasic collisions would be very helpful to me.

    I want a simple answer as I am not a mathematician to follow mathematical language.
  7. Dec 30, 2013 #6
    If that is the case, then How do we distinguish a scalar zero from a scalar vector?
  8. Dec 30, 2013 #7
    The vector has implied dimensionality. Zero scalar means 0 in 1 dimension. Zero vector means 0 in all components of the vector.
  9. Dec 31, 2013 #8
    First of all, sorry if I was too technical with the answer, I just thought I'd provide some more details since the previous posters had already pretty much answered your original question. It's by no means essential, so just ignore it if you want.

    The difference is that they are, well, completely different objects. Scalars are somehting you use to multiply vectors with, whereas vectors are typically [Unless you introduce new operations] added together. The zero scalar in R3 is 0, the zero vector is (0,0,0).

    What do you mean by saying that "it is no different from a scalar zero"? The (lack of) set direction for a zero vector is irrelevant, you can already see that it is a different object from the zero scalar because you can add it to any other vector, but you can't add a vector and the zero scalar together.

    This is probably the most direct definition of both: When you multiply any vector by the zero scalar, you get the zero vector, but when you add the zero vector to any vector, you get the original vector.

    Unfortunately I can't think of any way to explain the difference between the zero vector and the zero scalar by relying on binary collisions.

    (Btw, if you ever plan on studying quantum mechanics, for example, you're very likely to encounter more abstract vectors and the definition of a general vector space I mentioned earlier, they are not "just for mathematicians", but also used in physics a lot.)
  10. Dec 31, 2013 #9

    Let us say, we have two equal and opposite vectos parallel to X-axis. Their sum is a zero vector.

    Now we consider two equal and opposite vectors in an arbitrary direction. Their sum is a zero vector.

    Are the above two zero vectors equal?
  11. Dec 31, 2013 #10
    As I said, there is only one zero vector in a vector space. So yes, for any vector the sum of it and its additive inverse (the 'equal and opposite vector') is the same zero vector. To perhaps make it more transparent, let [itex]\mathbf{v},\mathbf{w}[/itex] be two (different) vectors. Then [itex]\mathbf{v}+(-\mathbf{v})=\mathbf{0}=\mathbf{w}+(-\mathbf{w})[/itex] Because that is how you define what [itex]-\mathbf{v}[/itex] means.

    [This is probably irrelevant for you, so feel free to ignore it, but for completeness, here's a proof why the zero vectors are unique. Let [itex]\mathbf{0},\mathbf{0'}[/itex] be zero vectors. Remember that zero vector fulfills [itex]\mathbf{v}+\mathbf{0}=\mathbf{v}[/itex] for every vector by definition. So [itex]\mathbf{0}=\mathbf{0}+\mathbf{0'}[/itex] because [itex]\mathbf{0'}[/itex] is a zero vector. But addition of vectors is commutative, so we may switch the order to say [itex]\mathbf{0}=\mathbf{0'}+\mathbf{0}[/itex]. But the RHS is just [itex]\mathbf{0'}[/itex], since [itex]\mathbf{0}[/itex] is also a zero vector. So [itex]\mathbf{0}=\mathbf{0}+\mathbf{0'}=\mathbf{0'}+\mathbf{0}=\mathbf{0'}[/itex])
    Last edited: Dec 31, 2013
  12. Dec 31, 2013 #11
    Can we resolve zero vector into two equal and opposite vectors of arbitrary magnitude in any direction of our choice?

    Is it possible to compare two zero vectors? If yes, how do we do that?
  13. Dec 31, 2013 #12
    Let us say V and W have different directions. Then the magnitude of the sum of V and -V, and the magnitude of the sum of W and -W are equal. But how can we can we equate their directions since V and W have different directions? And, we know two vectors are said to be equal if and only if they both have the same direction and magnitude.

    Please understnad these are my genuine problems and not for just for argument sake.
  14. Dec 31, 2013 #13
    Remember that the direction of the vector can change when you add it to another vector, so the direction of a sum of two vectors does not necessarily have to be the direction of either of the initial vectors.

    Sure, V,W may have different directions, but because the sums V+(-V), W+(-W) are both the zero vector, they have no particular direction attached to them.
  15. Dec 31, 2013 #14
    When we add two vectors we get a unique resultant vector with a definite magnitude and direction. Such being the case, V +(-V) has a direction parallel to the direction of V. This direction is different from the direction of the resultant vector W +(-W. Hence the question: How can V + (-V) be equal to W +(-Wwhen V and W are non colinear?
  16. Dec 31, 2013 #15
    This seems to be your main problem here. The zero vector simply does not have a set direction. An alternative convention is that it points to every direction, if that helps you.

    The "definition" of vectors as things "that have both magnitude and direction" can not only cause inconsistencies in things like this, it is also difficult (I'm tempted to say impossible) to generalize outside "regular" vectors of Rn.
  17. Dec 31, 2013 #16


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    Many of the folks posting on this thread have an understanding of vectors based on linear algebra and "vector spaces". When you first learn about vectors in high school and first year college physics, the presentation is entirely different.

    Given a presentation in which vectors are considered to consist of a magnitude and a direction, the zero vector is special. It has no direction. In such a presentation, two vectors are equal if either

    1: They have the same magnitude and direction.
    2: They both have zero magnitude.

    Note that the "polar coordinates" presentation that you are using is not incompatible with the linear algebra presentation. The linear algebra presentation is more general. Vectors considered as a magnitude and a direction with the usual addition and scalar multiplication operations fulfil the axioms of a vector space. [As do vectors considered as n-tuples of coordinate values].

  18. Dec 31, 2013 #17


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    it's true that V +(-V) is a scalar multiple of V and is also a scalar multiple of W +(-W) which is also a scalar multiple of W. But this scalar multiple is the zero scalar. you should not be surprised that 0*V + 0*W = 0 even though V and W are linearly independent.
  19. Dec 31, 2013 #18
    Yes, it is my main problem.

    If zero vector does not have a set direction it becomes an ill defined vector. On the other hand, on the alternative convention, if it points in every direction, then it looses uniqueness.

    Either way it gives rise to problems in conceptulising a zero vector. If we impose deliberately ceratin properties to zero vector, they could lead to inconsistencies in usage in various contexts.

    You would perhaps better appreciate my problem in the context of elastic collision analysis.

    In center of mass (CM) reference frame, a system of two masses has zero momentum before collision and it keeps on changing direction as the process of collision progresses in time. So momentum conservation which demands the momentum be zero at any instant of time is accounted for, with infinite number of zero vectors with different directions.

    In my openion a paradox is escaping scrutiny under the pretext of 'all zero vectors are equal'.
    Last edited by a moderator: Dec 31, 2013
  20. Dec 31, 2013 #19


    Staff: Mentor

    The zero vector is perfectly well defined and unique. It is defined under the axioms of a vector space. Heuristically vectors may be described as little arrows with a magnitude and a direction, but that is merely a descriptive analogy. The actual definition of a vector has to do with the axioms regarding vector addition and scalar multiplication. The directionality of a vector is not a defining feature.

    You simply need to let go of the misconception you have about how vectors are defined. The zero vector is not only perfectly well defined, it is an essential part of a vector space. Please pay attention to DeIdeal's excellent posts in this thread.
    Last edited: Dec 31, 2013
  21. Jan 1, 2014 #20


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    it looks like your problem is not with the zero vector. to me, it looks like your problem is that you are expecting to be able to know the individual momenta, if you are given the total momenta. But this is definitely not true. the individual momenta can be expressed as two vectors ##(P_{1x},P_{1y},P_{1z})## and ##(P_{2x},P_{2y},P_{2z})## And the total momentum can be expressed as just one vector ##(P_{x},P_{y},P_{z})##. So if I tell you only the total momentum, then of course you can't guess what are the individual momenta.
  22. Jan 1, 2014 #21

    D H

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    No, you have two main problems. One is your persistent claim that there are a multiplicity of zero vectors when you have been shown that there is only one. See DeIdeal's post #10.

    You other main problem is that you are using the rather naive definition of a vector as something that has magnitude and direction. This is not an essential characteristic of what it means for something to be a "vector". On the other hand, the existence of the zero vector is an essential characteristic. The zero vector is provably unique. Once again, see post #10.
  23. Jan 1, 2014 #22


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    Vectors are arrows attached to an origin, and the zero vector is the origin. You can think of the zero vector as having a special rule to define it as a vector.

    Then the only questions are whether such a definition is consistent, and whether it is useful. I don't know of a proof of consistency, so someone else will have to answer that, but it certainly has been useful.
  24. Jan 1, 2014 #23


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    You could just as well be talking about the number zero, as was mentioned in the first reply to your original post. It is neither positive or negative, yet is a perfectly well-defined number.

    Or (according to post #3 of this thread), it is defined by some to be both positive and negative, but is still a unique number.

    It seems like you are letting semantics get in the way of understanding a conceptually simple idea.
  25. Jan 1, 2014 #24


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    I was rather jokingly referring to:

    It's more of a engineering workaround, for limited accuracy in digital number representations. In math as such, zero has no sign. And the zero vector has no direction.
  26. Jan 1, 2014 #25


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    And as the author of post #2, I knew what you were doing :smile:.

    We also have the CDC 6x00 series machines of 40 years ago, which used one's-complement integer arithmetic and so had positive and negative representations of integer(!) zero... Picked up an entire clock cycle through the ALU that way, which is a big deal at 10 MHz.
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