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Normal (geometry) Line or vector perpendicular to a curve or a surface. A polygon and its two normal vectors. A normal to a surface at a point is the same as a normal to the tangent plane to the surface at the same point. In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object.
Illustration of tangential and normal components of a vector to a surface. In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the normal component of the vector.
However, normal may also refer to the magnitude of a vector. In particular, a set is called orthonormal (orthogonal plus normal) if it is an orthogonal set of unit vectors. As a result, use of the term normal to mean "orthogonal" is often avoided. The word "normal" also has a different meaning in probability and statistics.
The concept of an orthogonal basis is applicable to a vector space (over any field) equipped with a symmetric bilinear form , where orthogonality of two vectors and means . For an orthogonal basis : where is a quadratic form associated with (in an inner product space, ). Hence for an orthogonal basis , where ...
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a ...
The Frenet–Serret formulas were generalized to higher-dimensional Euclidean spaces by Camille Jordan in 1874. Suppose that r(s) is a smooth curve in and that the first n derivatives of r are linearly independent. [2] The vectors in the Frenet–Serret frame are an orthonormal basis constructed by applying the Gram-Schmidt process to the ...
Orthonormality. Property of two or more vectors that are orthogonal and of unit length. In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized.
Orthogonalization. In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly independent set of vectors {v1, ... , vk} in an inner product space (most commonly the Euclidean space Rn), orthogonalization results in a set of orthogonal vectors {u1 ...