Surface
More formally, let be a surface, and be a point on the surface. Let be a vector at . Then one can write uniquely as a sum
where the first vector in the sum is the tangential component and the second one is the normal component. It follows immediately that these two vectors are perpendicular to each other.
To calculate the tangential and normal components, consider a unit normal to the surface, that is, a unit vector perpendicular to at . Then,
and thus
where "" denotes the dot product. Another formula for the tangential component is
where "" denotes the cross product.
These formulas do not depend on the particular unit normal used (there exist two unit normals to any surface at a given point, pointing in opposite directions, so one of the unit normals is the negative of the other one).