# 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.
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 $$\vec{OP_2} = \vec V_2$$ have 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)