View source for Direction cosine

{{Short description|Cosines of the angles between a vector and the coordinate axes}}
{{no footnotes|date=January 2017}}

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 positive coordinate axes. Equivalently, they are the contributions of each component of the [[Basis_(linear_algebra)|basis]] to a [[unit vector]] in that [[direction (geometry)|direction]].

==Three-dimensional Cartesian coordinates ==

[[File:Direction cosine vector.svg|thumb|Vector {{math|'''v'''}} in {{tmath|\R^3}}]]
[[File:Direction cosine unit vector.svg|thumb|Direction cosines and direction angles for the unit vector <math>\tfrac{\mathbf v}{| \mathbf v|}</math>]]
{{further|Cartesian coordinates}}

If {{math|'''v'''}} is a [[Euclidean vector]] in [[Three-dimensional space|three-dimensional]] [[Euclidean space]], {{tmath|\R^3,}}

<math display=block>\mathbf v = v_x \mathbf e_x + v_y \mathbf e_y + v_z \mathbf e_z,</math>

where {{math|'''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 display=block>\begin{alignat}{2}
 \alpha &{}= \cos a = \frac{\mathbf v \cdot \mathbf e_x}{\Vert\mathbf v\Vert } &&{}= \frac{v_x}{\sqrt{v_x^2 + v_y^2 + v_z^2}},\\
 \beta  &{}= \cos b = \frac{\mathbf v \cdot \mathbf e_y}{\Vert\mathbf v\Vert } &&{}= \frac{v_y}{\sqrt{v_x^2 + v_y^2 + v_z^2}},\\
 \gamma &{}= \cos c = \frac{\mathbf v \cdot \mathbf e_z}{\Vert\mathbf v\Vert } &&{}= \frac{v_z}{\sqrt{v_x^2 + v_y^2 + v_z^2}}.
\end{alignat}</math>

It follows that by squaring each equation and adding the results

<math display=block> \cos^2 a + \cos^2 b + \cos^2 c = \alpha^{2} + \beta^{2} + \gamma^{2} = 1.</math>

Here {{math|''α'', ''β'', ''γ''}} are the direction cosines and the Cartesian coordinates of the [[unit vector]] <math>\tfrac{\mathbf v}{| \mathbf v|},</math> and {{math|''a'', ''b'', ''c''}} are the direction angles of the vector {{math|'''v'''}}.

The direction angles {{math|''a'', ''b'', ''c''}} are [[acute angle|acute]] or [[obtuse angle]]s, i.e., {{math|0 ≤ ''a'' ≤ π}}, {{math|0 ≤ ''b'' ≤ ''π''}} and {{math|0 ≤ ''c'' ≤ ''π''}}, and they denote the angles formed between {{math|'''v'''}} and the unit basis vectors {{math|'''e'''<sub>''x''</sub>, '''e'''<sub>''y''</sub>, '''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#Conversion_between_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. Simply put, direction cosines provide an easy method of representing the direction of a vector in a Cartesian coordinate system.

== Applications ==

=== Determining angles between two vectors ===
If vectors {{math|'''u'''}} and {{math|'''v'''}} have direction cosines {{math|(''α<sub>u</sub>'', ''β<sub>u</sub>'', ''γ<sub>u</sub>'')}} and {{math|(''α<sub>v</sub>'', ''β<sub>v</sub>'', ''γ<sub>v</sub>'')}} respectively, with an angle {{mvar|θ}} between them, their units vectors are

<math display=block>\begin{align}
  \mathbf{\hat u} &= \frac{1}{\sqrt{u_x^2 + u_y^2 + u_z^2}}\left(u_x \mathbf e_x + u_y \mathbf e_y + u_z \mathbf e_z\right) = \alpha_u \mathbf e_x + \beta_u \mathbf e_y + \gamma_u \mathbf e_z \\
  \mathbf{\hat v} &= \frac{1}{\sqrt{v_x^2 + v_y^2 + v_z^2}}\left(v_x \mathbf e_x + v_y \mathbf e_y + v_z \mathbf e_z\right) = \alpha_v \mathbf e_x + \beta_v \mathbf e_y + \gamma_v \mathbf e_z.
\end{align}</math>
Taking the [[dot product]] of these two unit vectors yield,
<math display=block>
  \mathbf{\hat u \cdot \hat v} = \alpha_u\alpha_v + \beta_u\beta_v + \gamma_u\gamma_v = \cos \theta,
</math>
where {{mvar|θ}} is the angle between the two unit vectors, and is also the angle between {{math|'''u'''}} and {{math|'''v'''}}.

Since {{mvar|θ}} is a geometric angle, and is never negative. Therefore only the positive value of the dot product is taken, yielding us the final result,

<math display=block>\theta = \arccos \left(\alpha_u\alpha_v + \beta_u\beta_v + \gamma_u\gamma_v\right).</math>

==See also==

* [[Cartesian tensor]]
* [[Euler angles]]

==References==

*{{cite book |first=D. C. |last=Kay |title=Tensor Calculus| series=Schaum’s Outlines|publisher=McGraw Hill|pages=18–19| year=1988 | isbn=0-07-033484-6}}
*{{cite book |edition=2nd |first1=M. R. |last1=Spiegel |first2=S. |last2=Lipschutz |first3=D. |last3=Spellman | title=Vector analysis| series=Schaum’s Outlines|publisher=McGraw Hill |pages=15, 25| year=2009 | isbn=978-0-07-161545-7}}
*{{cite book |title=An introduction to tensor analysis for engineers and applied scientists |first=J. R. |last=Tyldesley |publisher=Longman |page=5 |year=1975 |isbn=0-582-44355-5 |url=https://books.google.com/books?id=PODXAAAAMAAJ }}
*{{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|url=http://mathworld.wolfram.com/DirectionCosine.html}}[[Category:Algebraic geometry]]
[[Category:Vectors (mathematics and physics)]]


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