How to Write Vectors in Spherical Coordinates for Scalar Product Evaluation

  • Context: Undergrad 
  • Thread starter Thread starter Ki-nana18
  • Start date Start date
  • Tags Tags
    Angle Vectors
Click For Summary
SUMMARY

The discussion focuses on deriving the angle ϒ between two vectors defined in spherical coordinates, specifically using the relation cos ϒ = cosΘ1*cos Θ2 + sinΘ1*sinΘ2*cos(Φ1-Φ2). The Cartesian components of the vectors are expressed as x = ρ*sinΘ*cosΦ, y = ρ*sinΘ*sinΦ, and z = ρ*cosΘ. The challenge lies in converting these spherical components into the vector form B = Bx x-hat + By y-hat + Bz z-hat for evaluation of the scalar product.

PREREQUISITES
  • Spherical coordinates and their definitions
  • Vector representation in Cartesian coordinates
  • Understanding of scalar products in vector mathematics
  • Trigonometric identities related to angles
NEXT STEPS
  • Study the conversion of spherical coordinates to Cartesian coordinates in detail
  • Learn about vector operations including addition and scalar products
  • Explore trigonometric identities and their applications in vector mathematics
  • Practice problems involving angles between vectors in various coordinate systems
USEFUL FOR

This discussion is beneficial for students and professionals in physics, mathematics, and engineering, particularly those dealing with vector analysis and coordinate transformations.

Ki-nana18
Messages
90
Reaction score
0
Here is the problem verbatim:
The polar and azimuthal angles of a vector are Θ1 and Φ1. The polar and azimuthal angles of a second vector are Θ2 and Φ2. Show that the angle ϒ between the two vectors satisfies the relation:

cos ϒ = cosΘ1*cos Θ2+sinΘ1*sinΘ2*cos(Φ1-Φ2)

Hint: write out the Cartesian components of each vector in spherical coordinates and then evaluate the scalar product.

Where I run into trouble with this problem is when writing out the Cartesian components into Spherical components. I do know that x = ρ*sinΘ*cosΦ, y = ρ*sin Θ*sinΦ, and z=ρ cos Θ, but I'm not sure how to write a vector in the form of B= Bxx-hat+Byy-hat+Bzz-hat using the x, y, and z spherical components. Someone please point me in the right direction.
 
Physics news on Phys.org
I just realized I posted this in the wrong section of the forum. Sorry.
 

Similar threads

Graduate How to find the displacement vector in Spherical coordinate
  • · Replies 11 ·
Replies
11
Views
6K
Undergrad Dot product of two vector operators in unusual coordinates
  • · Replies 14 ·
Replies
14
Views
3K
Undergrad Transforming coordinates between Cartesian and spherical
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Question about the vector cross product in spherical or cylindrical coordinates
  • · Replies 5 ·
Replies
5
Views
5K
Undergrad Dot product in spherical coordinates
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Representing displacement vectors in cylindrical coordinates
  • · Replies 18 ·
Replies
18
Views
19K
High School To calculate the polar unit vectors using rotated coordinates
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Contravariant Vector Transformation in Spherical Polar Coordinates
  • · Replies 13 ·
Replies
13
Views
2K
High School Method of images and spherical coordinates
  • · Replies 4 ·
Replies
4
Views
2K
How to find the curl of a vector field which points in the theta direction?
  • · Replies 33 ·
2
Replies
33
Views
5K