In elementary geometry, orthogonal is the same as perpendicular. Two lines or curves are orthogonal if they are perpendicular at their point of intersection. Two vectors v and w of the real plane R^2 or the real space R^3 are orthogonal iff their dot product v·w=0. This condition has been exploited to define orthogonality in the more abstract context of the n-dimensional real space R^n.

More generally, two elements v and w of an inner product space E are called orthogonal if the inner product of v and w is 0. Two subspaces V and W of E are called orthogonal if every element of V is orthogonal to every element of W. The same definitions can be applied to any symmetric or differential k-form and to any Hermitian form.

See also

Group Orthogonality Theorem, Orthogonal Array, Orthogonal Basis, Orthogonal Circles, Orthogonal Complement, Orthogonal Coordinate System, Orthogonal Curves, Orthogonal Decomposition, Orthogonal Functions, Orthogonal Group, Orthogonal Group Representations, Orthogonal Involution, Orthogonal Lie Algebra, Orthogonal Lines, Orthogonal Matrix, Orthogonal Polynomials, Orthogonal Projection, Orthogonal Set, Orthogonal Subspaces, Orthogonal Sum, Orthogonal Surfaces, Orthogonal Tensors, Orthogonal Transformation, Orthogonal Vectors, Orthogonality Condition, Perpendicular

This entry contributed by Margherita Barile

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Barile, Margherita. "Orthogonal." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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