# Orthonormal, Orthogonal, Perpendicular

Nusc
What is the difference between these terms?
In what context do they apply to?
How important is it that we treat them differently?

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Nusc said:
What is the difference between these terms?
In what context do they apply to?
How important is it that we treat them differently?
When speaking of two vectors u,v perpendicular and orthoganal are used interchangably to mean that an inner product is zero.
<u|v>=0.
Perpendicular sometimes, but not always is used to indicate that the inner product in question has geometric interpitations. In that context <u|v> would mean two lines related to the vectors form right angles.
Orthoganal is applied to linearly independent sets to mean that for any two vectors in a set <v(i)|v(j)>=0 if i and j are not the same. Orthonormal means that in addition to being orthoganal <v(i)|v(i)>=1. This is quite use full because the problem of determining the coefficient of a vector in a representation of a vector by a basis is in general dependent on solving a linear system, but reduces in a orthoganal basis to finding an inner product.
because
v=a1v1+a2vi+...
so
<v(i)|v>=a(i)<v(i)|v(i)>
since the other terms are zero.

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Gold Member
orthogonal is from greek: ortho = right, gonal =angle. So I guess primarily, orthogonal was an adjective applying to lines: two lines at right angle with each other are said to be orthogonal. In this context, 'orthogonal' and 'perpendicular' are synomyms. This definition of orthogonality can easily be expanded to vectors in 2 or 3 dimensional space. Two vectors are orthogonal if their graphical representation (the arrows) are at 90° to one another. However, the concept of orthogonality has expended beyong the realm of 3 dimensional geometry. The algebraic (as opposed to geometric) condition for having two 3D vectors (a,b,c) and (d,e,f) to be orthogonal is that ad+be+cf = 0. So the generalisation of the notion of orthogonality to vectors of R^n is obvious: two vectors or R^n (a_1,...,a_n) and (b_1,...,b_n) are said to be orthogonal if a_1b_1+...+a_nb_n = 0. About orthonormality: two vectors are orthonormal if they are orthogonal and their norm is 1.

And the notion of orthogonality goes beyond that of vectors. For exemple, two functions f(x) and g(x) are said to be orthogonal over the interval [a,b] with weighting function w(x) if their inner product, defined as the integral of fgw from a to b, is 0. We also defined orthonormality between functions as "f(x) and g(x) are orthonormal over the interval [a,b] with weighting function w(x) if the integral of fgw from a to b, is 0 if f and g are not equal and is 1 if they are equal."

Note that you could have learned much of that by browsing on http://mathworld.wolfram.com/

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Orthogonal is essentially a generalisation of the word perpendicular.
Orthonormal means both orthogonal and normalized.

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unfortunately there is also one standard misuse of the term orthogonal in the case of "orthogonal group" O(n), referring to the group of n by n matrices whose columns form an orthonormal basis.