# How to find angle between two vectors, given their spherical co-ordinates?

• MHB
• WMDhamnekar
In summary, the conversation is about a formula involving arccos and two angles, and the question is asking for a proof of this formula. The conversation also includes a discussion about the Cartesian coordinates of two points and the use of inner product to prove a result involving these points.
WMDhamnekar
MHB

I know that $\arccos{(\cos{\phi_1}\cos{\phi_2}+\sin{\phi_1}\sin{\phi_2}\cos{(\theta_2-\theta_1)})}=\gamma$ But how can i answer the above question? If any member knows the proof of this formula may reply to this question with correct proof.

Last edited:
Dhamnekar Winod said:
https://www.physicsforums.com/attachments/10317

I know that $\arccos{(\cos{\phi_1}\cos{\phi_2}+\sin{\phi_1}\sin{\phi_2}\cos{(\theta_2-\theta_1)})}=\gamma$ But how can i answer the above question? If any member knows the proof of this formula may reply to this question with correct proof.
The Cartesian coordinates of $P_1$ are $(\rho_1\cos\theta_1, \rho_1\sin\theta_1\cos\phi_1,\rho_1\sin\theta_1\sin\phi_1)$, and similarly for $P_2$. Take the inner product, and use the fact that $\def\bv{\mathbf{v}} \langle\bv_1,\bv_2\rangle = |\bv_1||\bv_2|\cos\gamma.$

Opalg said:
The Cartesian coordinates of $P_1$ are $(\rho_1\cos\theta_1, \rho_1\sin\theta_1\cos\phi_1,\rho_1\sin\theta_1\sin\phi_1)$, and similarly for $P_2$. Take the inner product, and use the fact that $\def\bv{\mathbf{v}} \langle\bv_1,\bv_2\rangle = |\bv_1||\bv_2|\cos\gamma.$
In my opinion, the Cartesian co-ordinates of $P_1,P_2$ are as follows:

So your answer differs from the above answer for Cartesian co-ordinates of $P_1, P_2$ using appropriate $\rho_i, \theta_i,\phi_i$ where i=1,2.

Dhamnekar Winod said:
So your answer differs from the above answer for Cartesian co-ordinates of $P_1, P_2$ using appropriate $\rho_i, \theta_i,\phi_i$ where i=1,2.
The only difference is that I use $\theta$ and $\phi$ where you are using $\phi$ and $\theta$. If you switch the $\theta$s and $\phi$s in my hint then you should be able to prove the result.

@Opalg, Do you mean inner product=$\vec{v_1} \cdot \vec{v_2}=\left\langle v_1,v_2 \right\rangle$

Yes. :)

## 1. How do I convert spherical coordinates to Cartesian coordinates?

To convert spherical coordinates (r, θ, φ) to Cartesian coordinates (x, y, z), use the following formulas:
x = r * sin(θ) * cos(φ)
y = r * sin(θ) * sin(φ)
z = r * cos(θ)

## 2. How do I find the magnitude of a vector given its spherical coordinates?

The magnitude of a vector can be found using the formula:
|v| = √(x² + y² + z²)
where x, y, and z are the Cartesian coordinates obtained from converting the spherical coordinates.

## 3. What is the formula for finding the angle between two vectors in spherical coordinates?

The formula for finding the angle between two vectors in spherical coordinates is:
cos(θ) = (v1 * v2) / (|v1| * |v2|)
where v1 and v2 are the vectors and |v1| and |v2| are their magnitudes.

## 4. Can the angle between two vectors in spherical coordinates be negative?

No, the angle between two vectors in spherical coordinates is always positive. If the calculated angle is negative, it means that the vectors are pointing in opposite directions.

## 5. Is there a specific range for the angles in spherical coordinates?

Yes, the range for the angles in spherical coordinates is:
0° ≤ θ ≤ 180°
0° ≤ φ ≤ 360°
where θ is the polar angle and φ is the azimuthal angle.

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