Ângulos Diretores de Vetor 3D
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
cos α = x/|v|
About this calculator
The 3D vector direction angles calculator determines the angles a vector makes with each coordinate axis (X, Y, Z) in three-dimensional space. From the vector coordinates (x, y, z), it computes the vector magnitude (|v| = sqrt(x² + y² + z²)) and then the direction cosines: cos α = x/|v|, cos β = y/|v|, cos γ = z/|v|. The angles α, β, and γ are obtained via the arccosine of these values, in degrees or radians.
This tool is useful for students and professionals in engineering, physics, and mathematics who need to analyze vector orientation in problems of statics, dynamics, or analytic geometry. For example, when calculating the direction of a force or the position of an object in space, direction angles help understand the vector's tilt relative to the axes.
Important precautions: ensure coordinates are correct and the vector is not null (zero magnitude), as this would cause division by zero. Also, note that direction angles are unique for each vector, but two parallel vectors (same direction and sense) have the same angles. The calculator provides accurate results, but always check the selected angle unit.
Frequently asked questions
What are direction angles of a vector?
They are the angles the vector makes with the positive X, Y, and Z axes. They indicate the vector's orientation in 3D space.
How to calculate direction angles manually?
Compute the vector magnitude (square root of sum of squares of coordinates). Then divide each coordinate by the magnitude and apply arccosine to get each angle.
Can I use direction angles for 2D vectors?
Yes, but there are only two angles (with X and Y axes). The 3D calculator can be used by setting the z coordinate to zero.
What happens if the vector is zero?
A zero vector has no defined direction, so direction angles cannot be calculated. The calculator will return an error or undefined value.
Are direction angles unique for each vector?
Yes, each non-zero vector has a unique set of direction angles. However, parallel vectors with the same direction have the same angles.