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If you define a direction on a plane as either positive or negative, what would you call its positive/negativeness? Its orientation? I feel like there is a better word out there for this.

So like, if there is an arrow that starts at (0,0) and has a slope of 1, and there is another arrow that starts at (0,0) and has a slope of -1, would I say, the first arrow is the same as the second arrow but with opposite orientation?

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I would say direction. The second arrow has the same slope as the first, but the opposite direction.

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The word is sign:

In mathematics, the word sign refers to the property of being positive or negative. Every nonzero real number is either positive or negative, and therefore has a sign. Zero itself is signless, although in some contexts it makes sense to consider a signed zero. In addition to its application to real numbers, the word sign is used throughout mathematics to indicate aspects of mathematical objects that resemble positivity and negativity, such as the sign of a permutation.

On your arrows, if the gradients were +2 and -2 then they have opposite signs; if they were +2 and -0.5 they would be orthogonal or perpendicular.

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Sign is the mathematical term. –  Yuval Filmus Apr 21 '11 at 21:07

What you are describing here I think should be called a vector (as opposed to an "arrow"), so let's look at that definition from NOAD:

vector |ˈvektər| noun 1 Mathematics & Physics a quantity having direction as well as magnitude, esp. as determining the position of one point in space relative to another.

It seems pretty clear we're talking about direction here. But in your example your vectors are not congruent, so each has a different location. It wouldn't be enough to say they have different directions.

It is probably more accurate to say these are line segments with the same length and slope but different locations, or different start and end points, etc.

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If you're looking for something that specifies some direction along a single dimension (either positive or negative) then positivity might work.

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But then for negative directions, we would say "negative positivity," so awkward...! –  barf Jun 30 '11 at 7:44
    
@JTL: yeah, like 'negative growth'. I'm thinking positivity isn't that useful. –  Mitch Jun 30 '11 at 20:26

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