2D Vector Direction

The direction of a 2D vector $V = (x, y)$ is the angle $\theta$ it makes with the positive x-axis, measured counterclockwise.

$$ \theta = \operatorname{atan2}(y, x) $$

3D Vector Direction

The direction of a 3D vector $V = (x, y, z)$ is described by two angles:

• Azimuthal Angle ($\theta$): The angle in the xy-plane, same as the 2D case.
• Polar Angle ($\phi$): The angle from the positive z-axis.

$$ \theta = \operatorname{atan2}(y, x) $$ $$ \phi = \arccos\left(\frac{z}{\sqrt{x^2 + y^2 + z^2}}\right) $$