Orthogonal

Pronunciation: /ɔrˈθɒ gə nl/ ?
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Manipulative 1: Orthogonal lines. Created with GeoGebra.

Two lines are orthogonal if they are perpendicular at the point of intersection.[1] Two curves are orthogonal at a point of intersection if they are perpendicular at the point of intersection. Click on blue points in manipulatives 1 and 2 and drag them to change the figures. The objects in the figures remain orthogonal.

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Manipulative 2: Orthogonal curves. Created with GeoGebra.
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Manipulative 3: Orthogonal vectors. Created with GeoGebra.

Two vectors are orthogonal if the inner product of the vectors is zero. When graphing orthogonal vectors, the vectors are perpendicular to each other. Click on the blue points in manipulative 3 and drag them to change the figure. The vectors in manipulative 3 are always orthogonal.

Two square matrices of the same dimensions are orthogonal if the product of the matrices is the identity matrix. See figure 1.

[\\array{1 & 3 & -2 \\\\ 0 & -2 & 1 \\\\1 & 0 & 5}\\right]\\;\\cdot\\;\\left[\\array{\\frac{10}{11} & \\frac{15}{11} & \\frac{1}{22} \\\\ -\\frac{1}{11} & -\\frac{7}{11} & \\frac{1}{22} \\\\ -\\frac{2}{11} & -\\frac{3}{11} & \\frac{1}{11}}\\right]\\;=\\;\\left[\\array{1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1}\\right]
Figure 1: Orthogonal matrices

References

  1. orthogonal. http://wordnet.princeton.edu/. WordNet. Princeton University. (Accessed: 2011-01-08). http://wordnetweb.princeton.edu/perl/webwn?s=orthogonal&sub=Search+WordNet&o2=&o0=1&o7=&o5=&o1=1&o6=&o4=&o3=&h=.

Cite this article as:
Orthogonal. 2009-12-21. All Math Words Encyclopedia. Life is a Story Problem.org. http://www.allmathwords.org/en/o/orthogonal.html.

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2009-12-21: Added "References" (McAdams, David.)
2008-10-09: Initial version (McAdams, David.)

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