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On the example of a torus knot, the tangent vector T, the normal vector N, and the binormal vector B, along with the curvature κ(s), and the torsion τ(s) are displayed. At the peaks of the torsion function the rotation of the Frenet–Serret frame ( T , N , B ) around the tangent vector is clearly visible.
A real random vector = (, …,) is called a centered normal random vector if there exists a matrix such that has the same distribution as where is a standard normal random vector with components. [ 1 ] : p. 454
A normal vector of length one is called a unit normal vector. A curvature vector is a normal vector whose length is the curvature of the object. Multiplying a normal vector by −1 results in the opposite vector, which may be used for indicating sides (e.g., interior or exterior).
If the 4th component of the vector is 0 instead of 1, then only the vector's direction is reflected and its magnitude remains unchanged, as if it were mirrored through a parallel plane that passes through the origin. This is a useful property as it allows the transformation of both positional vectors and normal vectors with the same matrix.
All these extensions are also called normal or Gaussian laws, so a certain ambiguity in names exists. The multivariate normal distribution describes the Gaussian law in the k-dimensional Euclidean space. A vector X ∈ R k is multivariate-normally distributed if any linear combination of its components Σ k j=1 a j X j has a (univariate) normal ...
The magnitude of a vector (such as distance) is another example of an invariant, because it remains fixed even if geometrical vector components vary. (For example, for a position vector of length meters, if all Cartesian basis vectors are changed from meters in length to meters in length, the length of the position vector remains unchanged at ...
A random vector X ∈ R p (a p×1 "column vector") has a multivariate normal distribution with a nonsingular covariance matrix Σ precisely if Σ ∈ R p × p is a positive-definite matrix and the probability density function of X is
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.