Vector Direction Calculator

Calculator Form

Input Mode

Angle Unit

Decimal Precision

Vector Components

X Component

Y Component

Z Component

Point Coordinates

The vector is calculated as end point minus start point.

Start X

Start Y

Start Z

End X

End Y

End Z

Formula Used

The calculator supports direct vector components and two-point input. If you enter two points, it first builds the displacement vector.

For components: v = <x, y, z> For two points: v = <x2 - x1, y2 - y1, z2 - z1> Magnitude: |v| = sqrt(x² + y² + z²) Unit vector: u = v / |v| = <x/|v|, y/|v|, z/|v|> Direction cosines: cos(alpha) = x/|v| cos(beta) = y/|v| cos(gamma) = z/|v| Direction angles: alpha = arccos(x/|v|) beta = arccos(y/|v|) gamma = arccos(z/|v|) Azimuth in XY plane: theta = atan2(y, x) Elevation from XY plane: phi = atan2(z, sqrt(x² + y²))

These outputs describe the heading of the vector in space, its axis relationships, and its normalized directional form.

How to Use This Calculator

Choose Vector Components or Two Points.
Enter x, y, and z values directly, or enter start and end coordinates.
Select degrees or radians for angular output.
Choose the decimal precision you want.
Click Calculate Direction.
Read the vector, magnitude, unit vector, direction angles, azimuth, and elevation.
Use the CSV or PDF buttons to export the result.
Study the Plotly graph to visualize the final direction.

Example Data Table

Mode	Input	Vector	Magnitude	Azimuth (deg)	Elevation (deg)
Components	(3, 4, 0)	<3, 4, 0>	5.0000	53.1301	0.0000
Components	(2, -2, 1)	<2, -2, 1>	3.0000	315.0000	19.4712
Two Points	(1,2,0) to (4,6,0)	<3, 4, 0>	5.0000	53.1301	0.0000
Two Points	(0,0,1) to (2,1,5)	<2, 1, 4>	4.5826	26.5651	60.7941

Frequently Asked Questions

1. What does this calculator find?

It finds the direction of a vector using components or two coordinates. It also returns magnitude, unit vector, direction angles, azimuth, elevation, and directional cosines.

2. What is the difference between azimuth and elevation?

Azimuth measures the heading in the XY plane from the positive x-axis. Elevation measures how far the vector rises above or below the XY plane.

3. Why are there three direction angles?

A 3D vector can be compared with each coordinate axis. Alpha, beta, and gamma show the angles made with the positive x, y, and z axes.

4. What happens when I use two points?

The calculator subtracts the start point from the end point. That difference becomes the displacement vector used for every direction calculation.

5. Why can azimuth show N/A?

Azimuth depends on the horizontal projection. If both x and y are zero, the vector points only along z, so the XY heading is undefined.

6. What are directional cosines?

Directional cosines are the cosines of the angles between the vector and each axis. They match the unit vector components for a normalized vector.

7. Can this calculator work for 2D vectors?

Yes. Enter z as zero. The calculator will still compute magnitude and azimuth, while elevation becomes zero and the z-axis angle becomes ninety degrees.

8. What do the CSV and PDF downloads contain?

They export the current result summary, including vector form, magnitude, unit vector values, direction angles, azimuth, elevation, polar angle, and sign pattern.