Understanding Vectors vs Unit Vectors: Differences and Uses in Physics

[Ascendant78]

I am a bit confused about what the difference is between the two? To give some specific context where it has thrown me off, say if I were to define a charge with a vector r and compared that to a unit vector r hat, what exactly is the difference between what each of those tells me?

I have also seen it used for gravitational forces too, but I had never used the vector formula for it, so I don't know what that would tell me? It is odd to me because each of the formulas have an extra r in the denominator than the formula I was so used to using in Physics I for gravitational forces.

 

[grzz]

A UNIT vector has magnitude of 1.

If vector [itex]\underline{r}[/itex] has magnitude r then the unit vector in the direction of Vector [itex]\underline{r}[/itex] will be represented by [itex]\frac{\underline{r}}{r}[/itex].

 

[M Quack]

unit vectors always have length 1.

r hat is vector r divided by the length of vector r.

 

[grzz]

As M Quack said

The unit vector [itex]\hat{r}[/itex] is the vector [itex]\underline{r}[/itex] divided by the magnitude r of the vector [itex]\underline{r}[/itex] .

 

[Ascendant78]

Oh, I get their relation and the concept behind them in that a unit vector is 1 of whatever quantity is being measured. What I don't understand is exactly what they tell you relative to one another? Say if I found the vector force of gravity from a planet, then found the unit vector force of gravity from that same planet, I'm not sure of what each one would be telling me relative to each other? Like I said before, I only ever used the vector force of gravity, I never used the unit vector force, so I don't know what that would tell me about the gravitational force?

 

[Ascendant78]

Actually, I think I might get it now...

If I am understanding right, say if I calculated a unit vector force of a gravitational field between two objects to be say 500N and used meters for the units, then if they were 10m away from each other when I calculated this force, it means the force between them is 500N per m, making the overall force 5000N, or is that wrong?

 

[D H]

A UNIT vector has magnitude of 1.

And they are unitless. Each element of a position vector has units of length. Each element of an acceleration vector has units of length/time². A unit vector has no units.

 

[M Quack]

The force vector would give you the strength and direction of the force. The unit vector tells you only the direction.

 

[1MileCrash]

Actually, I think I might get it now...

If I am understanding right, say if I calculated a unit vector force of a gravitational field between two objects to be say 500N and used meters for the units, then if they were 10m away from each other when I calculated this force, it means the force between them is 500N per m, making the overall force 5000N, or is that wrong?

No, that doesn't make any sense at all.

Calculating a unit vector force doesn't mean anything. A unit vector is a dimensionless vector of length one. There is nothing to calculate. It is a vector of length one. It is not force, it is not Newtons, it is not anything. It is an indicator of "that way" and that is all.

Oh, I get their relation and the concept behind them in that a unit vector is 1 of whatever quantity is being measured. What I don't understand is exactly what they tell you relative to one another? Say if I found the vector force of gravity from a planet, then found the unit vector force of gravity from that same planet, I'm not sure of what each one would be telling me relative to each other? Like I said before, I only ever used the vector force of gravity, I never used the unit vector force, so I don't know what that would tell me about the gravitational force?

A unit vector is not a magnitude of 1 of "whatever it is you are measuring," it is just a magnitude of 1 in its direction. The "unit vector force of gravity" is something you've just.. invented, it doesn't mean anything.A vector for velocity expressed with unit vectors would look like:

V = 50 m/s i + 40 m/s j + 30 m/s k
Or
V = (50i + 40j + 30k) m/s

Where i, j, k are unit vectors. i, j, and k do not have a velocity magnitude 1. They merely indicate a direction. If i, j, k had a velocity magnitude 1, then the vector V's units would be m²/s², which is not what we are trying to say.In other words, if I told you to run 50 ft to the left, the unit vector is not "1 ft left" it is just "left."

I think all of your questions stem from thinking that unit vectors are just like physical vectors, but with a length of 1. That's not right. Unit vectors are merely indicators of some direction, they only point. They never represent any physical quantity alone.

 

[Ascendant78]

No, that doesn't make any sense at all.

Calculating a unit vector force doesn't mean anything. A unit vector is a dimensionless vector of length one. There is nothing to calculate. It is a vector of length one. It is not force, it is not Newtons, it is not anything. It is an indicator of "that way" and that is all.




A unit vector is not a magnitude of 1 of "whatever it is you are measuring," it is just a magnitude of 1 in its direction. The "unit vector force of gravity" is something you've just.. invented, it doesn't mean anything.


A vector for velocity expressed with unit vectors would look like:

V = 50 m/s i + 40 m/s j + 30 m/s k
Or
V = (50i + 40j + 30k) m/s

Where i, j, k are unit vectors. i, j, and k do not have a velocity magnitude 1. They merely indicate a direction. If i, j, k had a velocity magnitude 1, then the vector V's units would be m²/s², which is not what we are trying to say.


In other words, if I told you to run 50 ft to the left, the unit vector is not "1 ft left" it is just "left."

I think all of your questions stem from thinking that unit vectors are just like physical vectors, but with a length of 1. That's not right. Unit vectors are merely indicators of some direction, they only point. They never represent any physical quantity alone.

Oh, thank you so much for the clarification.

After looking the notes over more, I see now what was throwing me off. It is that the electric field surrounding an object is described with an equation that has a unit vector in it. They even use the unit vector when solving the problems involving charges related to multiple objects as well as problems finding locations where the sum of the charges is zero. I originally thought the equation was being used to solve direction and magnitude of the charge since it had k, q, and r in the equations. I also guess that them using unit vectors to solve for distances didn't really matter since all of the unit vectors end up cancelling out anyway. Nonetheless, the book is very convoluted and by no means does a decent job at explaining it.