Can You Subtract Vectors in Spherical Coordinates to Find Distance?

[FrogPad]

Hi all,

Would someone please re-enlighten me.

Say I have a vector in spherical coordinates:

$$\vec r_1 = \phi \hat{\phi} + \theta \hat{\theta} + R \hat{R}$$

Where $$r, \theta, R$$ are scalars and the corresponding hat notation is the unit vectors.

Say, I form a new vector $$r_2$$ in spherical coordinates.

Would the distance from r_1 to r_2 be given by the norm of r_2-r_1.


What I'm trying to ask is this:
1) In rectangular coordinates I can find the vector from one point to another, via V_ab = V_b - V_a
2) If I have two vectors in spherical coordinates, can I find the distance from one point to another with subtraction? Or do I need to convert the spherical vectors to rectangular, and then perform the subtraction.

 

[tiny-tim]

Hi FrogPad!

1) In rectangular coordinates I can find the vector from one point to another, via V_ab = V_b - V_a
2) If I have two vectors in spherical coordinates, can I find the distance from one point to another with subtraction? Or do I need to convert the spherical vectors to rectangular, and then perform the subtraction.

Your suspicion is correct … you certanily can't use subtraction.

Either convert to rectangular, or use the cosine rule:

r₁₂² = r₁² + r₂² - 2r₁r₂cosθ,

where in two dimensions θ = θ₁ - θ₂, but in three dimensions θ is a lot more complicated!