In 4-dimensional Euclidean space the orthogonal complement of a line is a hyperplane and vice versa and that of a plane is a plane. Orthogonal functions Main article: Orthogonal functions

It is common to use the following inner product for two functions f and g: Here we introduce a nonnegative weight function w(x) in the definition of this inner product. We say that those functions are orthogonal if that inner product is zero: We write the norms with respect to this inner product and the weight function as The members of a sequence fi : i 1 2 3 ... are: orthogonal on the interval ab if orthonormal on the interval ab if where is the Kronecker delta. In other words any two of them are orthogonal and the norm of each is 1 in the case of the orthonormal sequence. See in particular orthogonal polynomials. Examples The vectors (1 3 2) (3 1 0) (1/3 1 5/3) are orthogonal to each other since (1)(3) + (3)(1) + (2)(0) 0 (3)(1/3) + (1)(1) + (0)(5/3) 0 and (1)(1/3) + (3)(1) + (2)(5/3) 0. The vectors (1 0 1 0 ...)T and (0 1 0 1 ...)T are orthogonal to each other. Clearly the dot product of these vectors is 0. We can then make the obvious generalization to consider the vectors in Z2n: for some positive integer a and for 1 k a 1 these vectors are orthogonal for example (1 0 0 1 0 0 1 0)T (0 1 0 0 1 0 0 1)T (0 0 1 0 0 1 0 0)T are orthogonal. Take two quadratic functions 2t + 3 and 5t2 + t 17/9. These functions are orthogonal with respect to a unit weight function on the interval from 1 to 1. The product of these two functions is 10t3 + 17t2 7/9 t 17/3 and now The functions 1 sin(nx) cos(nx) : n 1 2 3 ... are orthogonal with respect to Lebesgue measure on the interval from 0 to 2. This fact is basic in the theory of Fourier series. Orthogonal polynomials Various eponymously named polynomial sequences are sequences of orthogonal polynomials. In particular: The Hermite polynomials are orthogonal with respect to the normal distribution with expected value 0. The Legendre polynomials are orthogonal with respect to the uniform distribution on the interval from 1 to 1. The Laguerre polynomials are orthogonal with respect to the exponential distribution. Somewhat more general Laguerre polynomial sequences are orthogonal with respect to gamma distributions. The Chebyshev polynomials of the first kind are orthogonal with respect to the measure The Chebyshev polynomials of the second kind are orthogonal with respect to the Wigner semicircle distribution. Orthogonal states in quantum mechanics In quantum mechanics two eigenstates of a Hermitian operator m and n are orthogonal if they correspond to different eigenvalues. This means in Dirac notation that unless m and n correspond to the same eigenvalue. This follows from the fact that Schrdinger's equation is a SturmLiouville equation (in Schrdinger's formulation) or that observables are given by hermitian operators (in Heisenberg's formulation). Art and architecture In art the perspective imagined lines pointing to the vanishing point are referred to as 'orthogonal lines'. The term "orthogonal line" often has a quite different meaning in the literature of modern art criticism. Many works by painters such as Piet Mondrian and Burgoyne Diller are noted for their exclusive use of "orthogonal lines" not however with reference to perspective but rather referring to lines which are straight and exclusively horizontal or vertical forming right angles where they intersect. For example an essay at the website of the Thyssen-Bornemisza Museum states that "Mondrian ....dedicated his entire oeuvre to the investigation of the balance between orthogonal lines and primary colours." 1 Computer science Orthogonality is a system design property facilitating feasibility and compactness of complex designs. Orthogonality guarantees that modifying the technical effect produced by a component of a system neither creates nor propagates side effects to other components of the system. The emergent behavior of a system consisting of components should be controlled strictly by formal definitions of its logic and not by side effects resulting from poor integration i.e. non-orthogonal design of modules and interfaces. Orthogonality reduces testing and development time because it is easier to verify designs that neither cause side effects nor depend on them. For example a car has orthogonal components and controls (e.g. accelerating the vehicle does not influence anything else but the components involved exclusively with the acceleration function). On the other hand a non-orthogonal design might have its steering influence its braking (e.g. electronic stability control) or its speed tweak its suspension.1 Consequently this usage is seen to be derived from the use of orthogonal in mathematics: One may project a vector onto a subspace by projecting it onto each member of a set of basis vectors separately and adding the projections if and only if the basis vectors are mutually orthogonal. An instruction set is said to be orthogonal if it lacks redundancy (i.e. there is only a single instruction that can be used to accomplish a given task)2 and is designed such that instructions can use any register in any addressing mode. This terminology results from considering an instruction as a vector whose components are the instruction fields. One field identifies the registers to be operated upon and another specifies the addressing mode. An orthogonal instruction set uniquely encodes all combinations of registers and addressing modes.citation needed Communications In communications multiple-access schemes are orthogonal when an ideal receiver can completely reject arbitrarily strong unwanted signals using different basis functions than the desired signal. One such scheme is TDMA where the orthogonal basis functions are non-overlapping rectangular pulses ("time slots"). Another scheme is orthogonal frequency-division multiplexing (OFDM) which refers to the use by a single transmitter of a set of frequency multiplexed signals with the exact minimum frequency spacing needed to make them orthogonal so that they do not interfere with each other. Well known examples include (a g and n) versions of 802.11 Wi-Fi; WiMAX; ITU-T G.hn DVB-T the terrestrial digital TV broadcast system used in most of the world outside North America; and DMT the standard form of ADSL. Statistics econometrics and economics When performing statistical analysis independent variables that affect a particular dependent variable are said to be orthogonal if they are uncorrelated.3 In this case the same results are obtained for the effect of any of the independent variables upon the dependent variable regardless of whether one models the variables' effects individually with simple regression or simultaneously with multiple regression. If correlation is present the factors are not orthogonal and different results are obtained by the two methods. This usage arises from the fact that if centered (by subtracting the expected value (the mean)) uncorrelated variables are orthogonal in the geometric sense discussed above both as observed data (i.e. vectors) and as random variables (i.e. density functions). One econometric formalism that is alternative to the maximum likelihood framework the Generalized Method of Moments relies on orthogonality conditions. In particular the Ordinary Least Squares estimator may be easily derived from an orthogonality condition between dependent variables and model residuals. Taxonomy In taxonomy an orthogonal classification is one in which no item is a member of more than one group that is the classifications are mutually exclusive. Combinatorics In combinatorics two nn Latin squares are said to be orthogonal if their superimposition yields all possible n2 combinations of entries. Chemistry In synthetic organic chemistry orthogonal protection is a strategy allowing the deprotection of functional groups independently of each other. System Reliability In the field of system reliability orthogonal redundancy is that form of redundancy where the form of backup device or method is completely different from the prone to error device or method. The failure mode of an orthogonally redundant backup device or method does not intersect with and is completely different from the failure mode of the device or method in need of redundancy to safeguard the total system against catastrophic failure. Neuroscience In neuroscience a sensory map in the brain which has overlapping stimulus coding (e.g. location and quality) is called an orthogonal map. See also Look up orthogonal in Wiktionary the free dictionary. Orthogonalization GramSchmidt process Orthogonal complement Orthonormality Pan-orthogonality occurs in coquaternions Orthonormal basis Orthogonal polynomials Orthogonal matrix Orthogonal group Surface normal Imaginary number Isogonal Isogonal trajectory References "Lincoln Mark VIII speed-sensitive suspension (MPEG video)". http://www.markviii.org/video/mark.mpg. Retrieved 2006-09-15.  Null Linda & Lobur Julia (2006). The essentials of computer organization and architecture (2nd ed.). Jones & Bartlett Learning. p. 257. ISBN 9780763737696. http://books.google.com/booksidQGPHAl9GE-IC&pgPA257.  Probability Random Variables and Stochastic Processes. McGraw-Hill. 2002. pp. 211. ISBN 0-07-366011-6.  Chapter 4 Compactness and Orthogonality in The Art of Unix Programming v d eTopics related to linear algebra Scalar   Vector  Vector space  Vector projection   Linear span  Linear map  Linear projection  Linear independence  Linear combination  Basis  Column space  Row space  Dual space  Orthogonality  Rank  Minor  Kernel (matrix)   Eigenvalues and eigenvectors   Least squares regressions  Outer product  Inner product space  Dot product  Transpose  GramSchmidt process  Matrix decomposition