Tangential and normal components

In mathematics and applications, given a vector to a surface at a point, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the surface, called the tangent component of the vector, and another one perpendicular to the surface, called the normal component of the vector.

More formally, let $S$ be a surface, and $x$ be a point on the surface. Let $\mathbf {v}$ be a vector at $x.$ Then one can write uniquely $\mathbf {v}$ as a sum

$\mathbf {v} =\mathbf {v} _{||}+\mathbf {v} _{\perp }$

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 ${\hat {n}}$ perpendicular to $S$ at $x.$ Then,

$\mathbf {v} _{\perp }=(\mathbf {v} \cdot {\hat {n}}){\hat {n}}$

and

$\mathbf {v} _{||}=\mathbf {v} -(\mathbf {v} \cdot {\hat {n}}){\hat {n}}.$

Another formula for the tangential component is

$\mathbf {v} _{||}=-{\hat {n}}\times ({\hat {n}}\times \mathbf {v} ),$

where $\times$ denotes the cross product.

Note that these formulas do not depend on the particular unit normal ${\hat {n}}$ used (at any point on the surface, there are two unit normals to it, pointing in the opposite direction, so one of them is the negative of the other one).

Vector identities can be used to show that the two expressions for the tangential component are equivalent.

References

• Rojansky, Vladimir (1979). Electromagnetic fields and waves. New York: Dover Publications. ISBN 0486638340.{{cite book}}: CS1 maint: publisher location (link)

• Benjamin Crowell (2003) Newtonian physics. (online version) ISBN 097046701X.