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A simple question : Can a point on a piece of paper represent a vector ? Can i say that a point "B" ( magnitude =0, because it's a point) , is having direction towards +x axis ?

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closed as off topic by Brandon Enright, Waffle's Crazy Peanut, akhmeteli, user1504, twistor59 Jun 8 '13 at 8:21

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@PeterKravchuk probably a high school teacher. Don't they preach that vectors are "values that possess both magnitude and direction"? It's understandable to see how some would take that and infer that all vectors have a direction –  Jim Jun 6 '13 at 15:08
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@PeterKravchuk any question is sensible if it is asked sincerely and with the genuine intent of learning from the answer –  Jim Jun 6 '13 at 15:25
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@VishwasGagrani, when I said about meaningless question, I did not mean to offend you or your question (just in case). What I mean is that it is meaningless to ask what is the direction of a null vector unless you specify what a direction is. –  Peter Kravchuk Jun 6 '13 at 15:26
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@Jim, I am sorry, I did not express my thought clearly. I mean a question not as something being asked, but as something that does or does not have an answer. In an abstract sense. –  Peter Kravchuk Jun 6 '13 at 15:29
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Any example of vector quantity, that has magnitude only, but no direction ? What i learnt is, if anything doesnot have a direction then it's scalar. –  Vishwas Gagrani Jun 6 '13 at 15:38
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5 Answers 5

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In Cartesian coordinates, a vector with magnitude 0 will not have a direction. However, in polar/spherical coordinates (or almost any system with angular coordinates), you could arbitrarily create a vector with r=0 but with $\theta$ or $\phi$ being some angle. Then it would technically have zero magnitude with an arbitrary direction, but this is both meaningless and, as John previously stated, it is not a point.

To reference your example, no, you cannot say point B has a direction toward the +x-axis because it has no component in that direction. The only way you can claim something has a certain direction is if its coordinates are such that everyone would agree on the direction. With your example, you could say point B is in +x direction, but I'd say it looks more like the -z direction. Since we can't agree, it can't have a direction.

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To expand on John Rennie's answer, the difference is a geometric difference: a point is not a vector, they are two distinct types of objects.

This distinction might be more clear by considering, for example, the inner product of a row and column vector in component form.

For a zero vector, the components of the vector are zero. But, the components exist and so, the inner product can be formed.

For a point, there are no components, no way to form the inner product with, e.g., a basis vector.

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You can have a null or zero vector, but this is not a point. It is an example of a zero tensor.

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For consistency, we would like (1) all vectors to have both a magnitude and a direction. We would also like (2) the sum of any two vectors to be a vector. Mathematicians a long time ago decided that 2 was more important than 1, so the zero vector is allowed into the club even though it has no direction. Mathematical systems like vector spaces, groups, fields, etc., are generally designed in a conservative way so that they are as much like the real number system as possible. Therefore we'd like vectors to have an additive identity, additive inverses, commutative addition, etc.

One way of thinking about the case of the zero vector is that in real life, numbers represent measurements, and every measurement has some uncertainty. Therefore when we're talking about a zero vector, it's not really zero and does have a direction, but we just don't know its direction with any decent precision.

Another approach is to define a vector as something that is defined by its transformation properties. For example, if I rotate my house by 180 degrees, its temperature stays the same, but the horizontal force of a broom leaning against the wall changes like $\textbf{F}\rightarrow-\textbf{F}$. This tells us that temperature is a scalar, but force is a vector.

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Ya, so as per the last paragraph, a quick litmus test to know, if a quantity is vector or scalar.. is to play with it's directions. If the output of the quantity changes with directions, it's vector! Otherwise, it's scalar –  Vishwas Gagrani Jun 6 '13 at 17:50
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A point may be viewed as zero magnitude vector with unspecified direction. I would argue that the direction is actually out of the plane.

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what if it's a 3-vector? –  Jim Jun 6 '13 at 14:58
    
Then out of the hyperplane?! –  ja72 Jun 6 '13 at 14:59
    
it would be out of space.. –  Arafat Jun 6 '13 at 14:59
    
you just blew my mind –  Jim Jun 6 '13 at 15:00
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