Understanding the Unit Vector: A Conceptual Explanation

[Ashley1nOnly]

Homework Statement

How does the unit vector have no units

I know that the unit vector has a length of 1 and zero units. A representation would be I-hat j-hat k-hat(depending on the coordinate system). But the unit vector is the vector/magnitude. If the unit vector is its vector/magnitude then how does it have a length of 1 and zero units.

Questions:
Is the unit vector the entire length of the vector(representation how much a vector points in each direction?
Does it only point towards the direction of the vector.
What exactly is the unit vector telling us.

Homework Equations

Just need an understanding of concepts

The Attempt at a Solution

Concept check

 

[Ashley1nOnly]

R-hat= r(vector)/ |r|
r=(x^2+y^2+z^2)
X/r i-hat+y/r j-hat+z/r k-hat

I am thinking that the X/r tell us how much(length) the vector is and the I hat tell us what direction the vector is pointed in.

The the vector is point in the I hat direction with a length of X/r

 

[PeroK]

Homework Statement

How does the unit vector have no units

I know that the unit vector has a length of 1 and zero units. A representation would be I-hat j-hat k-hat(depending on the coordinate system). But the unit vector is the vector/magnitude. If the unit vector is its vector/magnitude then how does it have a length of 1 and zero units.

Questions:
Is the unit vector the entire length of the vector(representation how much a vector points in each direction?
Does it only point towards the direction of the vector.
What exactly is the unit vector telling us.

Homework Equations

Just need an understanding of concepts

The Attempt at a Solution

Concept check

A unit vector is simply a vector of unit length. If you have any non-zero vector, $\vec{v}$, then you can find a unit vector in the same direction as $\vec{v}$. This is often written as $\hat{\vec{v}}$.

Unit vectors are useful useful as they separate the magnitude of the vector from its direction. For example, orthogonal unit vectors, like $\hat{i}, \hat{j}, \hat{k}$, are useful for expressing every vector as a set of three numbers $(x, y, z)$, from which it's relatively simple to do dot products, cross products and many other things.

Vectors as mathematical objects have no units.

 

[Ashley1nOnly]

R-hat = X I-hat + y j-hat +z k-hat
The magnitude of R is sqrt(x^2+y^2+z^2) which gives us the length of R

Now in order to find the length of each vector we need to dived by the magnitude of R.

Now the sum of all the vectors gives us the final vector R.
The I hat j hat k hat just let's us know what direction each vector component is point in.

And the r-hat unit vector is pointing in the r-hat direction after we have summed up all the vector component and got the final vector

in trying to understand what exactly each part of the equation is doing. Do I have the right understanding now?

 

[PeroK]

R-hat = X I-hat + y j-hat +z k-hat
The magnitude of R is sqrt(x^2+y^2+z^2) which gives us the length of R

Now in order to find the length of each vector we need to dived by the magnitude of R.

Now the sum of all the vectors gives us the final vector R.
The I hat j hat k hat just let's us know what direction each vector component is point in.

And the r-hat unit vector is pointing in the r-hat direction after we have summed up all the vector component and got the final vector

in trying to understand what exactly each part of the equation is doing. Do I have the right understanding now?

I'm not sure that you do understand. I would say that, if:

$R = x \hat{i} + y \hat{j} + z \hat{k}$

Then:

$\hat{R} = \frac{R}{|R|} = \frac{R}{\sqrt{x^2 + y^2 + z^2}} = \frac{x \hat{i} + y \hat{j} + z \hat{k}}{\sqrt{x^2 + y^2 + z^2}}$

 

[Ashley1nOnly]

So I went back and did so more knowledge digging.
The reason we say that the magnitude is sqrt(x^2+y^2+z^2) is because it's the same as when we were trying to fin the hypotenuse of a right triangle. (I never realized this before now) this find the length of our vector R. Which makes complete sense now.

Now our X I-hat + y j-hat + z k-hat our three vectors and when we add these three vectors we get our now vector point the the r-hat direction
Now r-hat is just telling us what direction the vector is pointing in.

I feel as though everything above is correct. Now after this needs some checking.

Now since we know the length of r-hat we would want to know how much of each vector component is contributing to the new vector which is why we divide by the overall length of r-hat.

 

[Ashley1nOnly]

So X/|r| is the length of the vector in the I-hat direction and so on. Now if we add all of the vectors together we get our final r-hat vector

 

[Buffu]

So X/|r| is the length of the vector in the I-hat direction and so on. Now if we add all of the vectors together we get our final r-hat vector

Is the unit vector the entire length of the vector(representation how much a vector points in each direction) ? - No
Does it only point towards the direction of the vector ? - Yes
What exactly is the unit vector telling us ? - Direction of vector.

Side note:- Why don't you learn latex to format maths. Maths is difficult to read and write for you and us, if not formatted.
Latex is not very difficult if you get the hang of it.

https://www.physicsforums.com/help/latexhelp/

 

[David Lewis]

If the unit vector is its vector/magnitude then how does it have a length of 1 and zero units?

When a vector represents a physical quantity, it has a unit of measure.
The magnitude of the vector also has the same unit of measure.
Unit vector = vector/magnitude of vector
The numerator and denominator have the same units so they cancel out.