Illustration of tangential and normal components of a vector 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 tangential component of the vector, and another one perpendicular to the surface, called the normal component of the vector. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... An open surface with X-, Y-, and Z-contours shown. ... In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ... Fig. ...

More formally, let S be a surface, and x be a point on the surface. Let be a vector at x. 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 S at x. Then, A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ... In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...

and

where "" denotes the dot product. Another formula for the tangential component is In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ...

where "" denotes the cross product. For the crossed product in algebra and functional analysis, see crossed product. ...

Note that 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).

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. In mathematics, a hypersurface is some kind of submanifold. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

References

• Rojansky, Vladimir (1979). Electromagnetic fields and waves. New York: Dover Publications. ISBN 0486638340. 

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