In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B are orthogonal when .

<geometry> At 90 degrees (right angles). N mutually orthogonal vectors span an N-dimensional vector space, meaning that, any vector in the space can be expressed as a linear combination of the vectors. This is true of any set of N linearly independent vectors. The term is used loosely to mean mutually independent or well separated. It is used to describe sets of primitives or capabilities that, like linearly independent vectors in geometry, span the entire "capability space" and are in some sense non-overlapping or mutually independent. For example, in logic, the set of operators "not" and "or" is described as orthogonal, but the set "nand", "or", and "not" is not (because any one of these can be expressed in terms of the others). Also used loosely to mean "irrelevant to", e.g. "This may be orthogonal to the discussion, but ...", similar to "going off at a tangent". See also orthogonal instruction set. [Jargon File] (2002-12-02)

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Orthogonal is a science fiction trilogy by Australian author Greg Egan taking place in a universe where, rather than three dimensions of space and one of time, there are four fundamentally identical dimensions. While the characters in the novels always perceive three of the dimensions as space and one as time, this classification depends entirely on their state of motion, and the dimension that one observer considers to be time can be seen as a purely spatial dimension by another observer.

In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.

In mathematics, orthogonal coordinates are defined as a set of d coordinates q = (q1, q2, ..., qd) in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents).

In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group.