stylistic:tone
Decomposition or resolution

As explained above, a vector is often described by a set of vector components that add up to form the given vector. Typically, these components are the projections of the vector on a set of mutually perpendicular reference axes (basis vectors). The vector is said to be decomposed or resolved with respect to that set.

The decomposition or resolution of a vector into components is not unique, because it depends on the choice of the axes on which the vector is projected.

Moreover, the use of Cartesian unit vectors such as x ^, y ^, z ^ {\displaystyle \mathbf {\hat {x}},\mathbf {\hat {y}},\mathbf {\hat {z}} } as a basis in which to represent a vector is not mandated. Vectors can also be expressed in terms of an arbitrary basis, including the unit vectors of a cylindrical coordinate system (ρ ^, ϕ ^, z ^ {\displaystyle {\boldsymbol {\hat {\rho }}},{\boldsymbol {\hat {\phi }}},\mathbf {\hat {z}} }) or spherical coordinate system (r ^, θ ^, ϕ ^ {\displaystyle \mathbf {\hat {r}},{\boldsymbol {\hat {\theta }}},{\boldsymbol {\hat {\phi }}}}). The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry, respectively.

The choice of a basis does not affect the properties of a vector or its behaviour under transformations.

A vector can also be broken up with respect to "non-fixed" basis vectors that change their orientation as a function of time or space. For example, a vector in three-dimensional space can be decomposed with respect to two axes, respectively normal, and tangent to a surface (see figure). Moreover, the radial and tangential components of a vector relate to the radius of rotation of an object. The former is parallel to the radius and the latter is orthogonal to it.

In these cases, each of the components may be in turn decomposed with respect to a fixed coordinate system or basis set (e.g., a global coordinate system, or inertial reference frame).