Understanding Vector Components and Negativity: A Scientific Explanation

[shredder666]

I asked my prof a question, and this is what he answered

"Vectors can't be negative; only components can be negative. (Only scalars can be negative, and components are scalars.) Vectors can be only zero or non-zero"

can someone explain this to me? I have the feeling that everything I've been taught was a lie D:

 

[Phrak]

By definition the norm of a vector is >=0. The norm corresponds to our notion of length. It would be perfectly fine to define the norm as <=0. It's just a definition, after all.

 

[Rasalhague]

If you think of a vector, v, as a directed line segment, an arrow, you can make it point in the opposite direction by multiplying it by -1, so -1 v = -v. This vector -v is the inverse of v, and v is the inverse of -v, meaning that -v + v = 0. But there's nothing inherently positive or negative about either. You could just as well have started with the vector I called -v, and name it w. Then -w = v. It's completely arbitrary which one you give the minus sign to, just a matter of how you label them. Given an arrow, there's no objective answer to the question "is it negative?" just as there's no objective answer to the question "which way is up?" in space.

Alternatively, if you think of a vector as a finite list real numbers called components, e.g. (x,y,z), with addition defined componentwise, (x,y,z) + (a,b,c) = (x+a,y+b,z+c), you could have some components negative, some positive, and some 0--they needn't all have the same sign--unless you're dealing with 1-dimensional vectors of this sort, in which case it'd be natural to call a negative number a negative vector. I guess you could invent a definition: let negative mean all components negative, but it's not a standard expression, as far as I know.

More generally, vectors are objects that obey certain axioms. For an arbitrary vector space, there's no standard definition of negative or positive vectors contained in these axioms.

 

[Danger]

As the uneducated member here, I must admit to being confused by this. Since velocity is a vector, would a deceleration not be considered a negative vector?
I'm not dismissing your response, Rasalhague... it is simply beyond my educational level. I can't understand it.

 

[Delta2]

I think your professor considers negativity only as a property of real numbers where a number x is negative iff x< 0. And if we want to be very accurate he is correct since there is no obvious ordering defined in the vectors so we can't speak of a vector x<0 (or x>0) .

So unless someone defines an ordering in the set of vectors he can't speak neither for positive or for negative vectors.

 

[Delta2]

As the uneducated member here, I must admit to being confused by this. Since velocity is a vector, would a deceleration not be considered a negative vector?
I'm not dismissing your response, Rasalhague... it is simply beyond my educational level. I can't understand it.

Deceleration and acceleration are vectors also. You could argue that deceleration is negative RELATIVE to the velocity vector since deceleration vector will have opposite direction RELATIVE to the velocity vector. BUT the direction of a vector cannot only be characterized as positive or negative (for example how would you characterize a vector whose direction is at 90 degrees angle with the velocity vector or another one at 45) and that's why i said it is not obvious to define an ordering in the set of vectors. Unless ofcourse
one considers only vectors of one dimension.

 

[Phrak]

You could just as well have started with the vector I called -v, and name it w. Then -w = v. It's completely arbitrary which one you give the minus sign to, just a matter of how you label them.

This part isn't correct. Both w and v have the same norm, length or magnitude--however you call it.

But, yeah, the statement "a vector isn't negative" is meaningless.

 

[Rasalhague]

This part isn't correct. Both w and v have the same norm, length or magnitude--however you call it.

But, yeah, the statement "a vector isn't negative" is meaningless.

Well, I took the question at face value as asking abut the possible negativity of vectors themselves. The OP didn't mention the norm of a vector, although you might be right in guessing that's what they meant.

 

[Phrak]

Well, I took the question at face value as asking abut the possible negativity of vectors themselves. The OP didn't mention the norm of a vector, although you might be right in guessing that's what they meant.

No worries. I'd thought it might be that, with how you are progressing so well through general relativity.

 

[Rasalhague]

As the uneducated member here, I must admit to being confused by this. Since velocity is a vector, would a deceleration not be considered a negative vector?
I'm not dismissing your response, Rasalhague... it is simply beyond my educational level. I can't understand it.

As a very slightly educated member... sorry!

I don't know if this will help, but suppose we have an object with a velocity of one unit in the x-direction of a Cartesian coordinate system, v = (1,0,0). Now suppose we decelerate this object till it's at rest by adding the vector -v = (-1,0,0).

v + (-v) = (1,0,0) + (-1,0,0) = 0.

Seems like (-1,0,0) is a good candidate to be called a "negative vector"? But what if our original object had been going in the opposite direction at the same speed. Then adding the same vector to the object's velocity results doesn't bring it to a standstill. It actually doubles its speed:

(-1,0,0) + (-1,0,0) = (-2,0,0).

The property of reducing speed, as opposed to increasing speed, isn't an inherent property of the vector, so we can't use it to define negativity in an absolute sense.

 

[Danger]

Thanks to both you and Delta for putting it into terms that were more understandable. It's still a tad out of my league, but at least I now have a bit of a handle on it.

 

[Studiot]

Funny that there is another thread currently ongoing about definitions involved with vectors.

Yes a vector can be negative, at least according to the vector space axiom 5.

if u, v & w are objects in a vector space V they are called vectors.

1) For every u, v in V there is a w = u + v

2) u + v = v + u

3) u + (v + w) = (u + v) + w

4) For every u in V there is an element, 0 in V such that u + 0 = 0 + u = u

5) For every u in V there is an object -u called the negative of u such that u + (-u) = (-u) + u = 0

6) If k is any scalar then ku is in V

7) k(u + v) = ku + kv

8) (k + m ) u = ku + mu

9) k(mu) = (km)(u)

10) 1u = u

 

[Delta2]

It is the negative of another vector or as i said earlier negative relative to another vector. To speak about negativity in the vectors in an absolute sense one has to define an ordering so he can compare every vector to the zero vector.

 

[shredder666]

OK all this axioms are just starting to confuse me, but so...
all in all its just a matter of convention?

 

[HallsofIvy]

Yes, you can have "the negative" of a vector but you cannot divide the set of all vectors into the 0 vector, "positive" vectors, and "negative" vectors. That is the point.