- #1

- 753

- 2

In what context do they apply to?

How important is it that we treat them differently?

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter Nusc
- Start date

- #1

- 753

- 2

In what context do they apply to?

How important is it that we treat them differently?

- #2

lurflurf

Homework Helper

- 2,440

- 138

When speaking of two vectors u,v perpendicular and orthoganal are used interchangably to mean that an inner product is zero.Nusc said:

In what context do they apply to?

How important is it that we treat them differently?

<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.

- #3

quasar987

Science Advisor

Homework Helper

Gold Member

- 4,784

- 18

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/

- #4

Galileo

Science Advisor

Homework Helper

- 1,991

- 6

Orthonormal means both orthogonal and normalized.

- #5

mathwonk

Science Advisor

Homework Helper

2020 Award

- 11,208

- 1,414

Share: