Vector decomposition

Vector decomposition is the decomposition of a vector of Rⁿ into several vectors, all linearly independent (in mutually distinct directions in the n-dimensional space).

Vector decomposition in two dimensions

In two dimensions, a vector can be decomposed in many ways. In the Cartesian coordinate system, the vector is decomposed into a portion along the ${\hat {x}}$ or ${\hat {i}}$ and the ${\hat {y}}$ or ${\hat {j}}$ directions.

One of the most common situations is when given a vector with magnitude and direction (or given in polar form), it can be converted into the sum of two perpendicular vectors (or converted to a Cartesian coordinate). In order to do this it makes use of trigonometry, such as sine and cosine.

Application in physics

Vector decomposition is used in physics to help adding vectors and hence solve many mechanical problems involving force, work, momentum, etc.

See also

• Coordinate system
• Helmholtz decomposition (decomposition of a vector field)