Gershgorin circle theorem

2013-11-11T11:59:48Z

202.121.182.12: /* Strengthening of the theorem */

In [[analytic geometry]], the '''direction cosines''' (or '''directional cosines''') of a [[Euclidean vector|vector]] are the [[cosine]]s of the angles between the vector and the three coordinate axes. Or equivalently it is the component contributions of the basis to the unit vector.

==Three dimensional Cartesian coordinates ==

[[File:Direction cosine vector.svg|thumb|Vector '''v''' in ℝ3.]]
[[File:Direction cosine unit vector.svg|thumb|Direction cosines and direction angles for the unit vector '''v'''/{{!}}'''v'''{{!}}.]]
{{further|Cartesian coordinates}}

If '''v''' is a [[Euclidean vector]] in [[Three-dimensional space|three dimensional]] [[Euclidean space]], ℝ3,

:<math>{\mathbf v}= v_\text{x} \mathbf{e}_\text{x} + v_\text{y} \mathbf{e}_\text{y} + v_\text{z} \mathbf{e}_\text{z}</math>

where '''e'''x, '''e'''y, '''e'''z are the [[standard basis]] in Cartesian notation, then the direction cosines are

:<math>\begin{align}
\alpha & = \cos a = \frac{{\mathbf v} \cdot \mathbf{e}_\text{x} }{ \left | {\mathbf v} \right | } & = \frac{v_\text{x}}{\sqrt{v_\text{x}^2 + v_\text{y}^2 + v_\text{z}^2}} ,\\
\beta & = \cos b = \frac{{\mathbf v} \cdot \mathbf{e}_\text{y} }{ \left | {\mathbf v} \right | } & = \frac{v_\text{y}}{\sqrt{v_\text{x}^2 + v_\text{y}^2 + v_\text{z}^2}} ,\\
\gamma &= \cos c = \frac{{\mathbf v} \cdot \mathbf{e}_\text{z} }{ \left | {\mathbf v} \right | } & = \frac{v_\text{z}}{\sqrt{v_\text{x}^2 + v_\text{y}^2 + v_\text{z}^2}}.
\end{align}
</math>

It follows that by squaring each equation and adding the results:

:<math> \cos ^2 a + \cos ^2 b + \cos ^2 c = 1\,.</math>

Here, ''α'', ''β'' and ''γ'' are the direction cosines and the Cartesian coordinates of the [[unit vector]] '''v'''/|'''v'''|, and ''a'', ''b'' and ''c'' are the direction angles of the vector '''v'''.

The direction angles ''a'', ''b'' and ''c'' are [[acute angle|acute]] or [[obtuse angle]]s, i.e., 0 ≤ ''a'' ≤ π, 0 ≤ ''b'' ≤ ''π'' and 0 ≤ ''c'' ≤ ''π'' and they denote the angles formed between '''v''' and the unit basis vectors, '''e'''x, '''e'''y and '''e'''z.

==General meaning==
More generally, '''direction cosine''' refers to the cosine of the angle between any two [[Euclidean vector|vector]]s. They are useful for forming [[Euclidean_vector#Multiple_Cartesian_bases|direction cosine matrices]] that express one set of [[orthonormal]] [[basis vectors]] in terms of another set, or for expressing a known [[Euclidean vector|vector]] in a different basis.

==See also==

* [[Cartesian tensor]]

==References==

*{{cite book| author=D. C. Kay| title=Tensor Calculus| series=Schaum’s Outlines|publisher=McGraw Hill|page=18-19| year=1988 | isbn=0-07-033484-6}}

*{{cite book|edition=2nd| author=M. R. Spiegel, S. Lipschutz, D. Spellman| title=Vector analysis| series=Schaum’s Outlines|publisher=McGraw Hill |page=15, 25| year=2009 | isbn=978-0-07-161545-7}}

*{{cite book|title=An introduction to tensor analysis for engineers and applied scientists
|author=J.R. Tyldesley|volume=|publisher=Longman|page=5|year=1975|series=|isbn=0-582-44355-5|url=http://books.google.co.uk/books/about/An_introduction_to_tensor_analysis_for_e.html?id=PODXAAAAMAAJ&redir_esc=y}}

*{{cite book |title=Mathematical Methods for Engineers and Scientists|volume=2|first=K.T.|last=Tang|publisher=Springer|year=2006|isbn=3-540-30268-9|page=13}}

*{{MathWorld|title=Direction Cosine|urlname=DirectionCosine|}}

[[Category:Algebraic geometry]]
[[Category:Vectors]]

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Gershgorin circle theorem