Vector Direction Calculator

Find the direction angle, azimuth, elevation, and direction cosines for any 2D or 3D vector.

Enter Vector Components

Result

Direction Angle
--
degrees
Vector--
Magnitude--
Direction Angle (from +x)--
Angle in Radians--
Compass Bearing--
Direction Cosines--
Unit Vector--

Step-by-Step Solution

theta = atan2(y, x)

Understanding Vector Direction

The direction of a vector describes where it points in space. In 2D, direction is specified by a single angle measured from the positive x-axis. In 3D, direction requires more information such as direction cosines, azimuth and elevation angles, or spherical coordinates.

Direction Measurements

Direction Angle (2D)

The angle from the positive x-axis to the vector, measured counter-clockwise. Range: -180 to 180 degrees.

theta = atan2(y, x)

Direction Cosines (3D)

The cosines of the angles between the vector and each coordinate axis. They satisfy cos^2(a) + cos^2(b) + cos^2(g) = 1.

cos(a)=x/|v|, cos(b)=y/|v|, cos(g)=z/|v|

Azimuth

The angle in the xy-plane measured from the positive x-axis. Used in navigation and spherical coordinates.

azimuth = atan2(y, x)

Elevation

The angle from the xy-plane to the vector. Positive means above the plane, negative means below.

elevation = atan2(z, sqrt(x^2+y^2))

Compass Bearing

Measured clockwise from north (positive y-axis). Common in navigation and surveying.

bearing = 90 - theta (adjusted to 0-360)

Polar/Spherical

Polar coordinates use (r, theta) in 2D. Spherical coordinates use (r, theta, phi) in 3D.

r = |v|, theta, phi

Applications

  • Navigation: Compass bearings for heading and course calculations.
  • Physics: Direction of forces, velocity vectors, electric fields.
  • Robotics: Orientation of robot arms, tool direction vectors.
  • Astronomy: Azimuth and elevation to locate celestial objects.
  • Game Development: Character facing direction, projectile trajectories.

Converting Between Representations

You can convert between Cartesian components, polar/spherical coordinates, and direction cosines. The key relationships are: x = |v| cos(alpha), y = |v| cos(beta), z = |v| cos(gamma). In 2D: x = |v| cos(theta), y = |v| sin(theta).