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Null vector / Zero vectorby Adjoint
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#1
Jun2514, 04:26 PM

P: 120

We know that a null vector (or zero vector) has direction but no magnitude. I am having some trouble understanding this concept.
1. What's the direction of a null vector, really? 2. Also as null vector is a vector, can velocity or acceleration also be null vectors? 3. If so, then what does a null velocity vector mean? Under which physical condition a particle can have such velocity? 4. Similarly, what does a null acceleration vector mean? Is there any physical example? I guess the example of a null position vector would be the position vector of a particle which is at the origin of the coordinate. Is this correct? I don't know if these have been asked before (couldn't find using the forum search). I appreciate your help. Thanks in advance. 


#2
Jun2514, 04:30 PM

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#3
Jun2514, 04:37 PM

P: 120

EDIT: By the way, I am an intro physics student. Just to mention my level. 


#4
Jun2514, 04:55 PM

Sci Advisor
P: 834

Null vector / Zero vector
In mathematics, a vector merely an element of a vector space. A vector space is merely a set with binary operations which satisfy the vector space axioms. It's a very abstract thing. 


#5
Jun2514, 04:57 PM

P: 1,302

The direction of a null vector is indeterminate, arbitrary. Any quantity that can be described with a vector has a null vector, it is a requirement, so yes, velocity, acceleration, all have null vectors.
A particle has such velocity whenever it is motionless relative to the frame from which it is being measured. A null acceleration vector means that the object is not accelerating. Yes, a null position vector would describe a "particle" at the origin of the coordinate system. It is just the zero of vectors. The analog that the number zero is neither positive nor negative should help you understand that the zero (or null) vector is not "directed" anywhere in a similar way. 


#7
Jun2514, 05:05 PM

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#8
Jun2514, 05:10 PM

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It makes more sense to me to teach the very basics of the concept of a vector space, then teach that you can think of the vectors you will see in the near future are things with a magnitude and a direction (except the zero vector). A vector space, conceptually, is a simple thing. The members of a vector space are "vectors". Only two operations are needed for a collection of objects to qualify as a vector space, vector addition and multiplication by a scalar. Each of these has concepts has multiple parts, but they just make sense. One aspect of addition is that the vector space has to have an additive identity, aka the "zero vector". One can get more abstract than that, but at the introductory level, that nicely covers the concept of a vector space. 


#9
Jun2514, 05:10 PM

P: 1,302

No, I do not mean that. The zero vector is not the same as the number zero. The null vector plays a similar role in the world of vectors as the number zero plays in the world of numbers. They are not the same thing. 


#10
Jun2514, 05:18 PM

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#11
Jun2514, 05:21 PM

P: 120

You said [itex]\vec{a}[/itex] = [itex]\vec{0}[/itex] means the object is not accelerating. But [itex]\vec{a}[/itex] = 0 also means that the object is not accelerating. What's the difference? 


#12
Jun2514, 05:51 PM

Sci Advisor
P: 834

Hence [itex]\vec{a}[/itex] = [itex]\vec{0}[/itex] is a true statement, but [itex]\vec{a}[/itex] = 0 is not (to be precise you can't take a vector and claim that it "equals" a scalar). 


#13
Jun2514, 05:57 PM

P: 120

I understand. Thanks everyone.
... One last question: What does indeterminate direction mean? Does it mean that it has no direction? Or does it mean we can't determine its direction (but it has one)? EDIT: I am asking this because if a null vector has no direction why not just say so? Why call it indeterminate? Also in the internet I found phrases such as null vector has no particular direction also null vector has every direction etc. 


#14
Jun2514, 06:11 PM

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The direction in which a vector ##\vec v## points is ##\vec v / \vec v##. In the case of the zero vector, this becomes ##\vec 0 / 0##. The analogy to 0/0 should be obvious. Another way to look at it: Two vectors ##\vec a## and ##\vec b## point in the same direction if there exists some scalar s such that ##\vec b = s\vec a##. For any vector ##\vec a##, scalar multiplication by zero yields the zero vector: ##0\vec a = \vec 0##. The analogy to 0*y=0 should be obvious. Either way you look at it, the direction in which the zero vector points is "indeterminate". 


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