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3D unit vectors

If a vector has a magnitude greater than that of i + j + k, for example i + 3j + 6k, What is represented by the unit vector of the vector? What makes it different from the unit vector of i + j + k?

I'm just struggling with what the unit vector is and means 🙂
A unit vector is a vector of length 1. The length of the vector ai + bj + ck is a2+b2+c2\sqrt{a^2+b^2+c^2}.

Note that the length of i + j + k is 3\sqrt{3}, so it isn't actually a unit vector. But 13(i+j+k)\dfrac{1}{\sqrt{3}}({\bf i} + {\bf j} + {\bf k}) is a unit vector.
(In general you can scale any non-zero vector to make it a unit vector - if has length L, just multiply your vector by 1/L).
Original post by DeFreedomWey
If a vector has a magnitude greater than that of i j k, for example i 3j 6k, What is represented by the unit vector of the vector? What makes it different from the unit vector of i j k?

I'm just struggling with what the unit vector is and means 🙂

The unit vector is just a vector which has a magnitude of one and usually you would need to attain it.

To find the unit vector you need to first find the magnitude. So let’s say you you have vector 2i 3j k.

To find the magnitude it’s each of the co-efficients of i, j and k squared added together and under the square root. Then you divide the initial vector equation by the magnitude. All what I said might be confusing so here’s a picture if that helps with all the steps. I have really good notes for vectors but they’re for IB, PM if you’re interested.

446A0F51-4211-444C-9436-88C9DF4C44A6.jpg.jpeg
(edited 4 years ago)
2C7F0659-B818-4253-89FA-51ADFA665814.jpg.jpeg25.4KB
Original post by JasmineLLB
The unit vector is just a vector which has a magnitude of one and usually you would need to attain it.

To find the unit vector you need to first find the magnitude. So let’s say you you have vector 2i 3j k.

To find the magnitude it’s each of the co-efficients of i, j and k squared added together and under the square root. Then you divide the initial vector equation by the magnitude. All what I said might be confusing so here’s a picture if that helps with all the steps. I have really good notes for vectors but they’re for IB, PM if you’re interested.

446A0F51-4211-444C-9436-88C9DF4C44A6.jpg.jpeg


Thank you, i know how to find the unit vector - i just don't know what it means 🙂
Original post by DeFreedomWey
Thank you, i know how to find the unit vector - i just don't know what it means 🙂


Ah ok my bad I didn’t read the full post 😅

You’re welcome :smile:
Original post by JasmineLLB
Ah ok my bad I didn’t read the full post 😅

You’re welcome :smile:


Please may you tell me what it means? 🙂
Original post by DeFreedomWey
Please may you tell me what it means? 🙂


Yeah I explained in my original post. I think to understand it you need to think of how you’re getting it.

I’m gonna send my notes:

B4A409F1-162F-4FF1-BFCC-DA16842371A6.jpg.jpeg
18E7459D-7412-483A-8245-022B805C9C52.jpg.jpeg
C52BF706-A3FC-4DD8-81CB-640779B4C7C3.jpg.jpeg
92223DF8-0387-4A1E-AEE4-564DE401E5B5.jpg.jpeg

If you still don’t get it, ask anything (:
(edited 4 years ago)
Original post by JasmineLLB
Yeah I explained in my original post. I think to understand it you need to think of how you’re getting it.

I’m gonna send my notes:

B4A409F1-162F-4FF1-BFCC-DA16842371A6.jpg.jpeg
18E7459D-7412-483A-8245-022B805C9C52.jpg.jpeg
C52BF706-A3FC-4DD8-81CB-640779B4C7C3.jpg.jpeg
92223DF8-0387-4A1E-AEE4-564DE401E5B5.jpg.jpeg


I'm very sorry but I'm finding it very hard to read the notes 😕.

If the vector i + j + 2k, the magnitude of it is root 6, is the unit vector of this vector a factor of the component of the resultant vector? 🙂
Original post by DeFreedomWey
I'm very sorry but I'm finding it very hard to read the notes 😕.

If the vector i + j + 2k, the magnitude of it is root 6, is the unit vector of this vector a factor of the component of the resultant vector? 🙂

I have no idea what you mean (or think you mean) by "a factor of the component of the resultant vector?" (I don't even know what you mean by "resultant vector" in fact).

Repeating myself: a unit vector is a vector of length 1.

If you have a general vector a\bf a, then the unit vector in the direction of a\bf a (typically written a^\bf \hat{a}) is given by dividing each component of a\bf a by the length of a\bf a (typically written ∣a∣|{\bf a}|).
Original post by DFranklin
I have no idea what you mean (or think you mean) by "a factor of the component of the resultant vector?" (I don't even know what you mean by "resultant vector" in fact).

Repeating myself: a unit vector is a vector of length 1.

If you have a general vector a\bf a, then the unit vector in the direction of a\bf a (typically written a^\bf \hat{a}) is given by dividing each component of a\bf a by the length of a\bf a (typically written ∣a∣|{\bf a}|).


Oops, I've just realised that I'm confusing the resultant vector with the magnitude of a vector, sorry about that 😕🙂.

Thank you for repeating the unit vector is a vector of length 1, i understand what it is now! 😄, although I'm not sure why dividing each of the components of the vector ( i,j,k) by the magnitude of the vector works to get you it🧐
Original post by DeFreedomWey
Oops, I've just realised that I'm confusing the resultant vector with the magnitude of a vector, sorry about that 😕🙂.

Thank you for repeating the unit vector is a vector of length 1, i understand what it is now! 😄, although I'm not sure why dividing each of the components of the vector ( i,j,k) by the magnitude of the vector works to get you it🧐

It's just scaling the vector so it has unit length.

e.g. 3i + 4j has length 5.
(3/5)i + (4/5)j has length 1.

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