Ângulo entre Vetores

The angle between vectors calculator determines the angle formed by two vectors in two-dimensional or three-dimensional space. The calculation uses the cosine formula: cos θ = (u·v) / (|u|·|v|), where u·v is the dot product and |u|, |v| are the magnitudes. The result is given in degrees or radians. This tool is useful for students and professionals in analytic geometry, physics, or computer graphics.

To use the calculator, enter the coordinates of the two vectors. For planar vectors, provide x and y components; for spatial vectors, add the z component. The tool automatically computes the dot product and magnitudes, then applies the cosine formula. The final angle is obtained via the arccosine function (acos). Ensure the vectors are non-zero, as division by zero is undefined.

Use cases include checking if two vectors are orthogonal (90° angle), parallel (0° or 180°), or determining the projection of one vector onto another. In physics, it calculates work done by a force, which depends on the angle between force and displacement. In computer graphics, it helps compute lighting and reflections. The calculator can also validate results from linear algebra exercises.

Cautions: ensure vectors are not zero. For higher dimensions, the formula extends, but this calculator supports only 2D and 3D. The arccosine result is between 0° and 180°. If cosine is greater than 1 or less than -1, rounding errors may occur; adjust values to the [-1, 1] interval.