Vector decomposition

Vector decomposition refers to decomposing a vector of Rⁿ into several vectors, each 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).

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)