Understanding Unit Vectors: A Step-by-Step Guide

In summary, the conversation is about finding the unit vector or normal vector in a specific form. One person asks for clarification on the manipulation that was used to get the vector in that form. Another person explains the derivation, which involves dividing by the magnitude of the vector. The conversation then shifts to finding the unit normal vector to a specific surface, with the final conclusion being that the unit vector can be written as \frac{x \vec{i}+y \vec{j}}{4}.
  • #1
JasonHathaway
115
0
Hi everyone,


Just want to know how does the the unit vector become in that form:

[itex]\vec{n}=\frac{2x\vec{i}+2y\vec{j}}{\sqrt{(2x)^{2}+(2y)^{2}}}=\frac{x \vec{i}+y \vec{j}}{4}[/itex]
 
Mathematics news on Phys.org
  • #2
Check your definition of "unit vector." :wink:
 
  • #3
As far as I know, the unit vector or the normal vector is the vector divided by its magnitude.

But that's not what I need to know, what I need to know is the manipulation that occurred.

[itex]\vec{n}=\frac{2x\vec{i}+2y\vec{j}}{\sqrt{(2x)^{2}+(2y)^{2}}}=\frac{2(x \vec{i}+y\vec{j})}{\sqrt{4(x^{2}+y^{2}})}=\frac{2(x \vec{i}+y\vec{j})}{2\sqrt{(x^{2}+y^{2}})}=\frac{x \vec{i}+y\vec{j}}{\sqrt{x^{2}+y^{2}}}[/itex]

That's my best. :Z
 
  • #4
JasonHathaway said:
As far as I know, the unit vector or the normal vector is the vector divided by its magnitude.

But that's not what I need to know, what I need to know is the manipulation that occurred.

[itex]\vec{n}=\frac{2x\vec{i}+2y\vec{j}}{\sqrt{(2x)^{2}+(2y)^{2}}}=\frac{2(x \vec{i}+y\vec{j})}{\sqrt{4(x^{2}+y^{2}})}=\frac{2(x \vec{i}+y\vec{j})}{2\sqrt{(x^{2}+y^{2}})}=\frac{x \vec{i}+y\vec{j}}{\sqrt{x^{2}+y^{2}}}[/itex]

That's my best. :Z

Derivation is correct.
 
  • #5
But how did it end up like this form: [itex]\frac{x \vec{i}+y \vec{j}}{4}[/itex]

And I've found something similar in Thomas Calculus:
wCpvMCp.jpg


Is [itex]y^{2} + z^{2}[/itex] equal to 1 or something? much like [itex]sin^{2}\theta + cos^{2}\theta = 1 [/itex]
 
  • #6
You're looking for "the" unit normal vector. Normal to what?
 
  • #7
Normal to the surface [itex]2x+3y+6z=12[/itex]
 
  • #8
Okay, but clearly that isn't where the gradient in the original post came from. So if you want to know what happened in post #3 (why x2 + y2 = 1) then you need to state the original problem.
 
  • #9
Sorry, that's not the correct surface, but the surface is [itex]x^{2}+y^{2}=16[/itex].
But I think I've got the idea:
[itex]\vec{n}=\frac{x\vec{i}+y \vec{j}}{\sqrt{x^{2}+y^{2}}}=\frac{x\vec{i}+y\vec{j}}{\sqrt{16}}=\frac{x\vec{i}+y\vec{j}}{4}[/itex]

right?
 
  • #10
:thumbs:
 

1. What is a unit vector?

A unit vector is a vector with a magnitude of 1 and is used to represent direction in a specific coordinate system. It is often denoted by a hat symbol (^) above the vector symbol.

2. How is a unit vector calculated?

To calculate a unit vector, divide a given vector by its magnitude. This will result in a vector with a magnitude of 1, making it a unit vector. Alternatively, you can also use the Pythagorean theorem to find the magnitude of a vector and then divide each component by the magnitude to get the unit vector.

3. What is the purpose of using unit vectors?

Unit vectors are used to represent direction in a specific coordinate system, making it easier to perform calculations and understand the direction of a vector. They also allow for easier comparison and manipulation of vectors in different coordinate systems.

4. Can a unit vector have a negative magnitude?

No, a unit vector by definition has a magnitude of 1 and cannot have a negative magnitude. However, the direction of a unit vector can be negative if it is pointing in the opposite direction of the coordinate system's positive direction.

5. How do unit vectors relate to the concept of basis vectors?

Unit vectors are often used as basis vectors in a coordinate system. In this case, they represent the fundamental directions (x, y, z) of the coordinate system. By scaling these unit vectors with their respective coordinates, any vector in the coordinate system can be represented.

Similar threads

  • General Math
Replies
16
Views
3K
  • General Math
Replies
1
Views
661
Replies
2
Views
1K
  • Linear and Abstract Algebra
Replies
3
Views
181
Replies
2
Views
198
  • Calculus and Beyond Homework Help
Replies
9
Views
699
  • Calculus and Beyond Homework Help
Replies
20
Views
384
  • Linear and Abstract Algebra
Replies
14
Views
497
  • Calculus and Beyond Homework Help
Replies
13
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
925
Back
Top