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.
That's exactly backwards. The zero vector has a magnitude (it's zero) but it has no direction (it's indeterminate).
Ok so zero vector has zero magnitude (I see that I mis-worded). But if it has no direction how is it a vector? EDIT: By the way, I am an intro physics student. Just to mention my level.
One of the things I see a lot teachers say is that the difference between vectors and scalars is that vectors have direction and magnitude while scalars only have magnitude. This is wrong on two fronts: the zero vector doesn't have direction, and if you choose complex numbers as your scalars then they have their own direction as well. If you were taught that, well its something you'll have to unlearn. 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.
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.
So do you mean [itex]\vec{v}[/itex] = [itex]\vec{0}[/itex] is same as [itex]\vec{v}[/itex] = 0? And similarly for acceleration? That is helpful! Thanks.
That's one of the reasons I don't like teaching that a vector is a thing with a magnitude and a direction. The zero vector has no direction. 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.
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.
I agree entirely. As a student, it really messed me up to go into vector analysis while thinking that vectors were "pointy line things."
Now here is some confusion for me. 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?
The difference is that [itex]\vec{0}[/itex] is a vector, but 0 is a scalar. 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).
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.
There's a subtle difference between undefined and indeterminate. I'll describe the difference in terms of dividing by zero. 1/0 is undefined, but 0/0 is indeterminate. The reason for the difference is how division is defined. a/b=c means that c is the number that makes b*c=a. So, can 1/0 have a value? If it does, call that value x. That means 0*x=1. But since 0*x=0 for all x, no such x exists. 1/0 is undefined. What about 0/0? Let's assume that exists, and call that y. That means 0*y=0. Since 0*y=0 for all y, the value of 0/0 can't be determined. It's "indeterminate". 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".
If you make the claim that the vector a is equal to the number 0, you made an utterly nonsensical statement. If you ever see that, the author is using the symbol "0" to refer to the zero vector.