Projeção de Vetor
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
proj escalar = u·v/|v|²
About this calculator
The vector projection calculator computes the projection of vector u onto vector v. The vector projection of u onto v is a vector in the direction of v whose length represents how much of u aligns with v. The formula used is proj_v(u) = (u·v / |v|²) · v, where u·v is the dot product and |v| is the magnitude of v. The result is a vector with the same direction as v (or opposite if the dot product is negative).
This tool is useful in various areas of mathematics and physics, such as force decomposition, computer graphics for shadow projection, and linear algebra for finding orthogonal components. For example, when analyzing a force on an inclined plane, the projection helps determine the component along the plane. It is also used in optimization and machine learning for dimensionality reduction.
When using the calculator, ensure that the vectors are in the same dimensional space (both 2D or 3D). The vector v cannot be zero, otherwise the projection is undefined. Note that the scalar projection (length) is u·v/|v|, while the vector projection includes direction. The calculator provides the resulting vector and, if desired, the magnitude of the projection.
Cautions: verify that the vector components are correct. The result may have decimal components; the calculator rounds to 4 decimal places. In physical problems, interpret the sign of the dot product: positive means the projection has the same direction as v; negative means opposite.
Frequently asked questions
What is vector projection?
It is the projection of one vector onto another, resulting in a vector in the direction of the second vector, whose length is the component of the first vector along that direction.
What is the difference between scalar and vector projection?
Scalar projection is the length (a number) of the projection, calculated by u·v/|v|. Vector projection is the resulting vector, which includes direction and sense.
Can I project a 2D vector onto a 3D vector?
No. The vectors must have the same dimension. The calculator accepts only 2D or 3D vectors, but both must be the same type.
What happens if vector v is zero?
The projection is undefined because the formula involves division by |v|². Make sure v is not the zero vector.
How to interpret the sign of the result?
If the dot product u·v is positive, the projection has the same direction as v. If negative, the projection points opposite to v.