Vector dot product in spherical co-ordinate

[clumps tim]

Hi, I am interested to see how to use the vector dot product formula in spherical coordinate system,

$$ V_1= r + \theta, at (1,0,0)$$ and $$ V_2= r - \theta, at (1, \frac{\pi}{2}, \frac{\pi}{2})$$

how to evaluate their dot product? do I have to transfer them into cartesian system? what would be the angle between them in cartesian system , please help

 

[jvicens]

Hello,

Thank you for your interest in using the vector dot product formula in a spherical coordinate system. To evaluate the dot product of two vectors in spherical coordinates, you do not necessarily have to transfer them into a Cartesian system. However, it may be easier to visualize and calculate the dot product in a Cartesian system.

To evaluate the dot product of two vectors, V1 and V2, in spherical coordinates, you can use the following formula:

V1 ⋅ V2 = (r1r2) cos(θ1) cos(θ2) + (r1r2) sin(θ1) sin(θ2) cos(ϕ1 - ϕ2)

In your case, V1 = r + θ at (1,0,0) and V2 = r - θ at (1, π/2, π/2). To evaluate their dot product, you would plug in the values of r, θ, and ϕ for each vector into the formula above.

Alternatively, if you would like to transfer the vectors into a Cartesian system, you can use the following conversion formulas:

x = r sin(θ) cos(ϕ)
y = r sin(θ) sin(ϕ)
z = r cos(θ)

Using these conversion formulas, you can transfer the vectors V1 and V2 into Cartesian coordinates and then calculate their dot product using the traditional formula:

V1 ⋅ V2 = x1x2 + y1y2 + z1z2

The angle between the two vectors in a Cartesian system can be calculated using the dot product formula as well:

cos(θ) = (V1 ⋅ V2) / (|V1| |V2|)

I hope this helps. Please let me know if you have any further questions. Good luck with your calculations!