# Angle Between two Vectors in Spherical Coordinates - Calculator

## Formulas Used in Calculations

Given two vectors by their spherical coordinates, this calculator calculates angle $\alpha$ between the two vectors..
Given two vectors whose initial point is the origin of a system of spherical coordintes and terminal points $P_1(\rho_1,\theta_1,\phi_1)$ and point $P_2(\rho_2,\theta_2,\phi_2)$ given by their spherical coordinates. Fig.1 - Angle $\alpha$ between two vetcors
We first convert the coordinates of points $P_1$ and $P_2$ into rectangular coordinates $P_1(x_1,y_1,z_1)$ and $P_2(x_2,y_2,z_2)$ where
$x_1 = \rho_1 \sin \phi_1 \cos \theta_1$ , $y_1 = \rho_1 \sin \phi_1 \sin \theta_1$ , $z_1= \rho_1 \cos \phi_1$
$x_2 = \rho_2 \sin \phi_2 \cos \theta_2$ , $y_2 = \rho_2 \sin \phi_2 \sin \theta_2$ , $z_2= \rho_2 \cos \phi_2$

The vectors $\vec{OP_1} = \vec V_1$ and $\vec{OP_2} = \vec V_2$ has the components
$\vec V_1 < x_1 , y_1 , z_1 >$ and $\vec V_2 < x_2 , y_2 , z_2 >$

The dot product of $\vec V_1$ and $\vec V_2$ is given by
$\vec V_1 \cdot \vec V_2 = ||\vec V_1 || \cdot ||\vec V_1 || \cos \alpha$
Hence
$\alpha = \arccos \left(\dfrac {\vec V_1 \cdot \vec V_2}{||\vec V_1 || \cdot ||\vec V_1 ||} \right)$
where
$\vec V_1 \cdot \vec V_2 = x_1 x_2 + y_1 y_2 + z_1 z_2$
and
$||\vec V_1 || = \sqrt {x_1^2 + y_1^2 + z_1^2}$ and $||\vec V_2 || = \sqrt {x_2^2 + y_2^2 + z_2^2}$
Note that if $||\vec V_1 || = 0$ or $||\vec V_2 || = 0$, the angle between the two vectors is undefined

## Use Calculator to Calculate Angle Bewteen two Vectors in Spherical Coordinates

1 - Enter the spherical coordinates $\rho_1$ , $\theta_1$, $\phi_1$ of point $P_1$, and the spherical coordinates $\rho_2$ , $\theta_2$, $\phi_2$ of point $P_2$, selecting the desired units for the angles, and press the button "Calculate". You may also change the number of decimal places as needed; it has to be a positive integer.

 $\rho_1 =$ 1 $\theta1 =$ 45 degrees radians $\phi_1 =$ 45 degrees radians $\rho_2 =$ 2 $\theta2 =$ 90 degrees radians $\phi_2 =$ 30 degrees radians
Number of Decimal Places =

$\alpha =$ (degrees)
$\alpha =$ (radians)