Tangential and normal components

Illustration of tangential and normal components to a surface.

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

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

where "$\cdot$" denotes the dot product. 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 (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).

Analogously, one defines the concepts of tangential and normal components of a vector to a curve in a plane, and to a $n-1$-dimensional hypersurface in a $n$-dimensional Euclidean space. One can still compute the tangential and normal components using the dot product, but the formula involving the cross product does not hold any more, since the cross product is defined only in three dimensions.

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.