Direction cosine#Cartesian coordinates

File:Direction cosine vector.svg

File:Direction cosine unit vector.svg

{{further|Cartesian coordinates}}

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

$$\backslash mathbf\; v\; =\; v\_x\; \backslash mathbf\; e\_x\; +\; v\_y\; \backslash mathbf\; e\_y\; +\; v\_z\; \backslash mathbf\; e\_z,$$

where {{math|e_x, e_y, e_z}} are the standard basis in Cartesian notation, then the direction cosines are

$$\backslash begin\{alignat\}\{2\}$$

\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}

It follows that by squaring each equation and adding the results

$$\backslash cos^2\; a\; +\; \backslash cos^2\; b\; +\; \backslash cos^2\; c\; =\; \backslash alpha^\{2\}\; +\; \backslash beta^\{2\}\; +\; \backslash gamma^\{2\}\; =\; 1.$$

Here {{math|α, β, γ}} are the direction cosines and the Cartesian coordinates of the unit vector $\backslash tfrac\{\backslash mathbf\; v\}$

The direction angles {{math|a, b, c}} are acute or obtuse angles, 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_x, e_y, e_z}}.

More generally, direction cosine refers to the cosine of the angle between any two vectors. They are useful for forming direction cosine matrices that express one set of orthonormal basis vectors in terms of another set, or for expressing a known 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.

If vectors {{math|u}} and {{math|v}} have direction cosines {{math|(α_u, β_u, γ_u)}} and {{math|(α_v, β_v, γ_v)}} respectively, with an angle {{mvar|θ}} between them, their units vectors are

$$\backslash begin\{align\}$$

\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}

Taking the dot product of these two unit vectors yield,

\mathbf{\hat u \cdot \hat v} = \alpha_u\alpha_v + \beta_u\beta_v + \gamma_u\gamma_v = \cos \theta,

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,

$$\backslash theta\; =\; \backslash arccos\; \backslash left(\backslash alpha\_u\backslash alpha\_v\; +\; \backslash beta\_u\backslash beta\_v\; +\; \backslash gamma\_u\backslash gamma\_v\backslash right).$$

• Cartesian tensor
• Euler angles

• {{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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