Advanced Vector Direction Angles Calculator

Calculator settings

Calculation mode

Angle unit

Angle range

Decimal precision

Use scientific notation when useful

Two-dimensional vector

x component

y component

Three-dimensional vector

x component

y component

z component

Two vectors

Vector a: x

Vector a: y

Vector a: z

Vector b: x

Vector b: y

Vector b: z

Initial and terminal points

Initial x

Initial y

Initial z

Terminal x

Terminal y

Terminal z

Magnitude and 2D direction

Magnitude

Direction angle

Magnitude, azimuth, and elevation

Magnitude

Azimuth

Elevation

Direction cosines and magnitude

Magnitude

l = cos α

m = cos β

n = cos γ

Normalize invalid cosine triplets

Projection of vector a onto vector b

Vector a: x

Vector a: y

Vector a: z

Vector b: x

Vector b: y

Vector b: z

Cross product vectors

Vector a: x

Vector a: y

Vector a: z

Vector b: x

Vector b: y

Vector b: z

Three-by-three matrix transformation

Vector x

Vector y

Vector z

Matrix entries

Euler rotation

Vector x

Vector y

Vector z

Rotation around x

Rotation around y

Rotation around z

Rotations are applied in X, then Y, then Z order.

Batch vector input

One vector per line

Use commas, spaces, or semicolons. Each line may contain two or three components.

Formula reference

2D direction angle

θ = atan2(y, x)

The two-argument arctangent preserves the correct quadrant.

Vector magnitude

|v| = √(x² + y² + z²)

Magnitude is always nonnegative.

3D direction angles

α = cos⁻¹(x/|v|), β = cos⁻¹(y/|v|), γ = cos⁻¹(z/|v|)

These angles are measured from the positive coordinate axes.

Angle between vectors

θ = cos⁻¹((a · b)/(|a||b|))

The result ranges from zero to a straight angle.

Projection

proj_b(a) = ((a · b)/|b|²)b

Projection measures the portion of one vector along another.

Azimuth and elevation

x = r cos e cos a, y = r cos e sin a, z = r sin e

Azimuth turns horizontally while elevation tilts vertically.

How to use this calculator

Select the calculation mode matching your known data.
Choose the preferred angle unit, range, and precision.
Enter vector components, points, angles, or matrix values.
Press Calculate to display results above the form.
Review formulas, graph output, validation, and JSON data.
Copy, print, share, or export the completed calculation.

Direction-angle conventions

A mathematical direction angle is measured counterclockwise from the positive x-axis. A navigation bearing is measured clockwise from north. Azimuth usually describes horizontal rotation, while elevation describes vertical tilt.

Three-dimensional coordinate direction angles use the positive x, y, and z axes. Their direction cosines satisfy a unit-sphere identity. Signed output ranges can represent clockwise or negative angular displacement.

Common questions

Why is atan2 better than ordinary arctangent?

It uses both components and preserves the correct quadrant. It also handles vertical vectors without dividing by zero.

Why does the zero vector cause an error?

The zero vector has no unique direction. Its unit vector and direction angles are undefined.

Can the calculator use radians?

Yes. It also supports degrees, gradians, and degree-minute-second output.

What does a negative direction cosine mean?

It means the vector points partly opposite the corresponding positive axis.

How are parallel vectors detected?

The calculator checks whether the cross-product magnitude is near zero.

How are perpendicular vectors detected?

The calculator checks whether the dot product is near zero.