Question

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.

Answer 1

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).

or

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".