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A vector has both magnitude and direction wheras a scalar only magnitude. If you have given -a, where a is a real number, it is a scalar because it has no direction, just magnitude. However if you deine direction as follows:
- <-----------> +
so that a
--------> vector is possitive and and a
<-------- vector is negative
what then would you classify -a? a vector or scalar?
any two vectors along the same geometric axis are considered 'in the same direction'...the vectors you have drawn form one geometric axis...however, there is a disctinction between parallel and anti parallel...parallel means you get one vector by multiplying another vector by a positive scalar...anti parallel means you get one vector by multiplying another vector by a negative scalar...but they're still both technically part of the same geometric axis.
http://mathworld.wolfram.com/Vector.html
My guess is that these things you define are vectors in R, I guess. Although there's very little distinction between these 1-dimensional vectors and the ordinary real number system...
Well, thanks for the read Happy_Reaper, it was good review. And to BobMcGee123; I never knew what anti-parallel meant.
>>what then would you classify -a? a vector or scalar?
However, I can't seem to to pick out the subliminal answer you guys have given me.
-a is a scalar that isn't useful in the context of vectors. You are incorrectly associating two ideas.
In the context of vectors, your assertion was correct that the scalar is only magnitude, whereas a vector has both magnitude and direction.
Subsequently, you are asking a question that doesn't make any sense. '-a' is, *strictly speaking* just a scalar, as you correctly mentioned, but when you try applying it to vectors it gets a little fuzzy.
1 - This is vector P
---------->
2 - This is vector P * 2
-------------------->
3 - This is the negative of vector P
<----------
4 - This is the negative of vector P, * 2
<--------------------
So far, I've broken this down into ideas that make sense. It is mathematically *correct* to say the vector in number 4 is just the vector in number 1 multiplied by negative 2, but intuitively that doesn't make any sense. Rather, it is only accepted to multiply a vector by a negative number when it's negative 1, and even then that's just equivalent to a 180 degree rotation.
The kicker here, as I mentioned above, is that a vector and it's 'anti-parallel' (or, the vector you get when multiplying each component by negative 1) still lay on the same geometric axis
This is a game of definitions. Once you see the ideas it's really not hard to grasp.
Actually, this morning, during my first period calculus class, it all came together! i feel so free! i wen't into physics and was adding two vectors, and my goodness! what a difference a comfort of a gut level understanding has on you!
Thanks Bob, for your input.