Tangential and normal components

Surface Tangential Component

The component of a vector tangential to a surface is the part of the vector that is directed along the surface. The tangential component of a vector field ${\vec {V}}$ can be calculated as

${\vec {V}}_{tang}=-{\hat {n}}\times ({\hat {n}}\times {\vec {V}}),$

where ${\hat {n}}$ is a unit surface normal vector. It does not matter which one of the two possible surface normals are used.

A more intuitive way of calculating the tangential component is to subtract the normal component from the vector:

${\vec {V}}_{tang}={\vec {V}}-({\vec {V}}\cdot {\hat {n}}){\hat {n}}.$

This can be done because the normal and tangential components are per definition orthogonal. Vector identities can be used to show that the two calculations are equivalent.