|
|
| Line 1: |
Line 1: |
| 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.
| | Greetings! I am Myrtle Shroyer. North Dakota is our birth place. I am a meter reader but I strategy on changing it. Doing ceramics is what my family members and I enjoy.<br><br>Feel free to surf to my blog [http://www.pathwayschico.org/blogs/post/216965 http://www.pathwayschico.org/blogs/post/216965] |
| | |
| ==Three dimensional Cartesian coordinates ==
| |
| | |
| [[File:Direction cosine vector.svg|thumb|Vector '''v''' in ℝ<sup>3</sup>.]]
| |
| [[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]], ℝ<sup>3</sup>,
| |
| | |
| :<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'''<sub>x</sub>, '''e'''<sub>y</sub>, '''e'''<sub>z</sub> 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'''<sub>x</sub>, '''e'''<sub>y</sub> and '''e'''<sub>z</sub>.
| |
| | |
| ==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]]
| |
Greetings! I am Myrtle Shroyer. North Dakota is our birth place. I am a meter reader but I strategy on changing it. Doing ceramics is what my family members and I enjoy.
Feel free to surf to my blog http://www.pathwayschico.org/blogs/post/216965