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Angle Between two Vectors in Cylindrical Coordinates - Calculator

Formulas Used in Calculations

An online calculator to calculate angle \( \alpha \) between these two vectors by their cylindrical coordinates is presented
Given two vectors whose initial point is the origin of a system of cylindrical coordinates and terminal points \( P_1(\rho_1,\theta_1,z_1) \) and \( P_2(\rho_2,\theta_2,z_2) \) given by their cylindrical coordinates.
angle between two vectors
Fig.1 - Angle \( \alpha\) between two vetcors

Convert the cylindrical coordinates of points \( P_1(\rho_1,\theta_1,z_1) \) and point \( P_2(\rho_2,\theta_2,z_2) \) into rectangular coordinates \( P_1(x_1,y_1,z_1) \) and \( P_2(x_2,y_2,z_2) \) where
\( x_1 = \rho_1 \cos \theta_1 \) , \( y_1 = \rho_1 \sin \theta_1 \) , \( z_1 = z_1\)
\( x_2 = \rho_2 \cos \theta_2 \) , \( y_2 = \rho_2 \sin \theta_2 \) , \( z_2 = z_2\)

The vectors \( \; \vec{OP_1} = \vec V_1 \) and \( \; \vec{OP_2} = \vec V_2 \) have the components
\( \vec V_1 \lt x_1 , y_1 , z_1 \gt \) and \( \; \vec V_2 \lt x_2 , y_2 , z_2 \gt \)

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 Cylindrical Coordinates

1 - Enter the cylindrical coordinates \( \rho_1 \) , \( \theta_1 \), \( z_1 \) of point \( P_1 \), and \( \rho_2\) , \( \theta_2\), \( z_2 \) 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.

\( \rho_1 = \)
\( \theta1 = \)
\( z_1 = \) 		\( \rho_2 = \)
\( \theta2 = \)
\( z_2 = \)
Number of Decimal Places =


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



More References and links

  1. Maths Calculators and Solvers.
  2. Convert Cylindrical to Rectangular Coordinates - Calculator .