# negative vectors (eg velocity)

If you said someone had a velocity of -12mph and they were traveling north? Wouldn't it mean that they were traveling 12mph south?

This is a quote from here:

if something [object-x] moving to the right was taken to have positive momentum, then one should consider something [object-y] moving to the left to have negative momentum.

But isn't Momentum a vector, so the direction should be specified separately to the number. What I mean is, object-y with -1200kg m/s and object-x with 1200kg m/s should both have momentum in the same direction. But this can't be because as the quote says they are moving in opposite direction.

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The word "speed" is usually used to denote the magnitude of the velocity. So you can't say that someone has a speed of -12 mph. The magnitude is always positive. – Greg P Jan 13 '11 at 20:21
now I understand why most HS kids find physics confusing. They might feel better knowing that the idiots writing their books are even more confused than they are. – Jeremy Jan 14 '11 at 14:08
Hey I dont think anyones an idiot. I'd just rather understand it the just learn "this happens, remember it, write it in the exam, forget it" and this forces side of things is my worst sub topic/area of physics. I prefer astrophysics, although I'm not saying the 2 are unrelated. Besides the point of this site is to increase your understanding and knowledge (and after that it acts as a reference which people use through search engines). I'm very sorry that I don't know as much as you but I'd like to get there. – Jonathan. Jan 14 '11 at 16:12
I have this problem with my students every quarter. – Carl Brannen Feb 19 '11 at 0:19

That quote is abit misleading, momentum is a vector, however a vector is neither negative nor positive, only its components can have this characteristic. The two objects you are describing does not have the same momentum, but they have the same magnitdue of momentum (length of vector).

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A vector is a combination of a force and a direction: youtube.com/watch?v=pXYlOC6u9w8 The speed in the example is not a vector unless it is known how much force is involved. – Steven Devijver Jan 13 '11 at 19:52
We need a downvote button on comments – user1708 Jan 13 '11 at 19:56
@kakemonsteret: just posting your own comment saying "no that's wrong" and getting it massively upvoted probably has about the same effect ;-) – David Z Jan 13 '11 at 21:35
@Steven: Completely incorrect. Many things that are not forces are vector quantities. To list a few obvious examples, velocity, momentum, position, electrical field... – Colin K Jan 13 '11 at 21:45
@Colin: except $\mathbf p$ is in fact a one-form, and electrical field is in general a $p$-form with values in Lie algebra (e.g. bosonic $\mathbf C$-field in 11d maximal supergravity). Also, position is not a vector unless you are working in a linear space(-time). But I guess we can stop playing these games :-) – Marek Jan 14 '11 at 3:14

There is nothing wrong with the quote because it assumes that the only allowed (or considered) motion is to the left or to the right. So the text is explaining things in the context of mechanics with one spatial dimension. And one-dimensional vectors are isomorphic to ordinary numbers. Their first and only component may be positive or negative, so one may also talk about positive and negative vectors.

Of course, this is not possible for higher-dimensional vectors. For at least 2-dimensional vectors, one has to talk about components with respect to specific axes if he wants to discuss the "signs of the momentum".

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In some sense you're right when you say "the direction should be specified separately to the number". However you have to be careful about what you mean--if you say this then by 'number' you mean an unsigned quantity (think absolute value). When you start talking about the signs of numbers, either positive or negative, then that sign encodes information about the direction that something is moving.

Note I have to define what 'positive' means and what 'negative' means, in the sense that positive doesn't always mean 'to the right'. It could mean 'to the left' as well. However once I make my choice I've set the meaning of positive and negative, and from then on 'positive' vectors point in that specific direction.

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Say I have 3 objects one going left and one going and one going downwards so they will each collide at the same point at the same time and have the same momentum. Then you can't use signs to represent direction. If you can't use signs then you should ever be allowed to use them which breaks the momentum thing as explained in the article. – Jonathan. Jan 14 '11 at 1:32