Derive the r, theta, and phi unit vectors?

In summary, the r, theta, and phi unit vectors are used in spherical coordinate systems to represent the magnitude and direction of a vector. They can be derived from the Cartesian unit vectors using trigonometric functions and have properties such as orthogonality and a magnitude of 1. They are specific to spherical coordinate systems and are related through a right-hand rule.
  • #1
Living_Dog
100
0
DJGriffiths, 3rd ed., Prob 1.37: derive the [tex]\hat{r}[/tex], [tex]\hat{\theta}[/tex], and [tex]\hat{\phi}[/tex] unit vectors in terms of [tex]\hat{x}[/tex], [tex]\hat{y}[/tex], and [tex]\hat{z}[/tex].

I know the formula and how to find them, but derive them?? ...unless this is what is meant?

tia,
-LD
 
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  • #2
I believe that is what it meant. Find them in terms of x,y,z.
 
  • #3
quasar987 said:
I believe that is what it meant. Find them in terms of x,y,z.

Ok! That's easy. Why Griffith used the word "derive" - as if from first principles - is beyond me.

thx, i needed that! :)

-LD
 
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1. What are the r, theta, and phi unit vectors used for?

The r, theta, and phi unit vectors are commonly used in spherical coordinate systems to represent a point in 3-dimensional space. They are used to describe the magnitude and direction of a vector relative to a given coordinate system.

2. How do you derive the r, theta, and phi unit vectors?

The r, theta, and phi unit vectors can be derived from the Cartesian unit vectors, i, j, and k, using trigonometric functions. The r unit vector is calculated as cos(theta)cos(phi)i + sin(theta)cos(phi)j + sin(phi)k. The theta unit vector is calculated as -sin(theta)i + cos(theta)j, and the phi unit vector is calculated as -sin(phi)cos(theta)i - sin(phi)sin(theta)j + cos(phi)k.

3. What are the properties of the r, theta, and phi unit vectors?

The r, theta, and phi unit vectors are orthogonal to each other, meaning they are perpendicular. They also have a magnitude of 1, making them unit vectors. Additionally, the theta unit vector is tangent to the latitude circle, and the phi unit vector is tangent to the longitude circle.

4. Can the r, theta, and phi unit vectors be used in any coordinate system?

No, the r, theta, and phi unit vectors are specific to spherical coordinate systems and are not interchangeable with unit vectors in other coordinate systems, such as Cartesian or cylindrical.

5. How are the r, theta, and phi unit vectors related to each other?

The r, theta, and phi unit vectors are related through a right-hand rule, where the r unit vector points in the direction of the radius, the theta unit vector points in the direction of increasing longitude, and the phi unit vector points in the direction of increasing latitude.

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