# if a vector has a magnitude equal to zero, can that thing exist? Can that thing be measured? [closed]

consider a vector (a vector is a physical quantity ) having magnitude equal to zero.Now, if something has a magnitude equal to zero, can that thing exist? Can that thing be measured? if net force on a particle is zero, then net force being a zero vector , will have some arbitrary direction ( according to definition of zero vector). is there any sense to say that the net force( in the above case) has some direction?
If the magnitude of the electric field is zero in a certain region of space , then there will not be any effect of electric field on a charge moving through that region. Thus , if there are not any physical effects of a quantity ,is this right to say that the physical quantity exists? If your answer is no , then how can we say that zero vector exists?

## closed as unclear what you're asking by user191954, AccidentalFourierTransform, John Rennie, Jon Custer, ZeroTheHeroNov 5 '18 at 2:12

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• Yes, you can measure zero. A caliper can start measuring a finite distance starting from 0 to any value within its calibrated range. – ja72 Nov 3 '18 at 16:56
• Zero itself is a measure. Its just a value and the existence of a value implies the existence of the very quantity that the value defines. – UKH Nov 3 '18 at 18:20
• The late physicist Stuart Freedman had a reputation for dealing rigorously with the question of how to approach the experimental measurement of quantities that might be (or even are expected to be) null. This is not an easy thing for non-trivial measurements. I seem to recall the phrase "measuring nothing and dong it right" floating around in the months following Stuart's death. – dmckee Nov 3 '18 at 21:54
• @ UKH,If the magnitude of the electric field is zero at somewhere in space, then there will not be any effect of electric field on a charge moving through that location. Thus , if there are not any physical effects of a quantity ,is this right to say that the physical quantity exists? – naveen Nov 4 '18 at 5:56
• @ ja72,If the magnitude of the electric field is zero at somewhere in space, then there will not be any effect of electric field on a charge moving through that location. Thus , if there are not any physical effects of a quantity ,is this right to say that the physical quantity exists? – naveen 7 hours ago – naveen Nov 4 '18 at 12:58

If something has a magnitude equal to zero, can that thing exist?

Consider the electric field. It seems to me that if the magnitude of the electric field is zero, it is still the case that the electric field exists. Otherwise, to what would the value of zero be assigned?

That is

$$|\vec{E}| = 0$$

is a statement about something that exists and not nothing.

• If the magnitude of the electric field is zero at somewhere in space, then there will not be any effect of electric field on a charge moving through that location. Thus , if there are not any physical effects of a quantity ,is this right to say that the physical quantity exists? – naveen Nov 4 '18 at 4:40
• @navee, the very statement "the electric field is zero somewhere in space" makes presumes the existence of, among things, an electric field to refer to. – Alfred Centauri Nov 4 '18 at 12:41
• ,let suppose a place or a volume in space where the electric field is zero. Then what should we say about the problem? where am i getting wrong? – naveen Nov 4 '18 at 12:52
• @naveen I think what Alfred is getting at is that electric fields exist in general, even if they can be $0$ in certain regions of space. From what I can tell you are going the other way though. If the field is $0$ within some region of space, then can we say in that region of space there is no electric field? Do I understand your issue? – Aaron Stevens Nov 5 '18 at 3:24
• @AaronStevens, my issue is - are these two statements same -(1)magnitude of electric field is zero and (2) there is no existence of any electric field. – naveen Nov 5 '18 at 3:34

Assuming that you are OK with the existence of zero as a real number in the first place, there is no added discomfort in admitting a zero vector. To deny it because its direction is ambiguous is to commit a grave mistake: much like how you have negative reals and some sums therefore must come to zero, like $$3+2+(-5)$$, some vectors like $$(2,1),(-1,1),(-1,-2)$$ sum to $$(0,0)$$.

Just like how in removing zero from the reals, to save the closure property of addition, you would have to eliminate all of the negative numbers too, if you want to remove the zero vector from the space you need to only look in one quadrant of the plane or one octant of 3D space, the space $$\mathbb R^n$$ being forced to become the space $$(\mathbb R^+)^n$$, and to outlaw vector subtraction.

Outlawing vector subtraction is a mistake because it means you can no longer take derivatives of vector-valued quantities, so you can no longer define velocity given position, or acceleration given velocity. That is a huge cost to pay when the alternative is just to be happy with the idea that you cannot move east or west when you are standing at the North Pole.

One uses the method of approaching a limit, the "epsilon delta" mathematical method. The limit is approached incrementally and gets infinitessimaly close .There are various examples in the link.

As zero will just be approached then also the vector directions are sustained.

If the problem is started with the premise "equal to zero" then direction for vectors will be undefined , but the value zero is a legal real number which can be assigned to any field.