Orthogonal means perpendicular in elementary geometry. Two lines are orthogonal
if they are perpendicular. Two curves
with well-defined tangent lines are orthogonal at
an intersection if their tangent
lines there are perpendicular. Two vectors 
and 
in 
or 
, and more generally in the real
vector space 
,
are orthogonal iff their dot product 
.
More generally, two vectors 
and 
of an inner product space 
are orthogonal if their inner
product is 0. Two subspaces 
and 
of 
are orthogonal if every vector
of 
is orthogonal to every vector
of 
. The same definition, 
, applies in a vector space
equipped with a symmetric bilinear form,
an alternating bilinear form, or a Hermitian form 
(Conrad n.d., pp. 4-6).
Orthogonality for these more general forms can differ from inner product orthogonality. For an alternating
bilinear form, every vector is orthogonal to itself,
including every nonzero vector
(Conrad n.d., pp. 2 and 6).
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 Lines,
Orthogonal
Matrix,
Orthogonal Polynomials,
Orthogonal Projection,
Orthogonal
Set,
Orthogonal Subspaces,
Orthogonal
Sum,
Orthogonal Surfaces,
Orthogonal
Tensors,
Orthogonal Transformation,
Orthogonal Vectors,
Orthogonality
Condition,
Perpendicular
Portions of this entry contributed by Margherita
Barile
Explore with Wolfram|Alpha
References
Conrad, K. "Bilinear Forms." Undated notes, pp. 2 and 4-7. https://kconrad.math.uconn.edu/blurbs/linmultialg/bilinearform.pdf.Referenced
on Wolfram|Alpha
Orthogonal
Cite this as:
Weisstein, Eric W., with contributions by Margherita Barile. "Orthogonal." From MathWorld--A Wolfram
Resource. https://mathworld.wolfram.com/Orthogonal.html
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