Vector decomposition

# Vector decomposition

Discussion

Encyclopedia
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
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

, the vector is decomposed into a portion along the or and the or directions.

One of the most common situations is when given a vector with magnitude and direction (or given in polar
Polar coordinate system
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction....

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
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

, work, momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

, etc.

• coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

• Helmholtz decomposition
Helmholtz decomposition
In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational vector field and a...

(decomposition of a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

)