Direction cosine

Vector with the direction angles , , . ${\displaystyle {\vec {v}}}$ ${\displaystyle \alpha _{1}}$ ${\displaystyle \alpha _{2}}$ ${\displaystyle \alpha _{3}}$

In the vector calculation are direction cosines of a vector of the Euclidean space , the cosine of its directional angle, ie the angle between the vector and the three standard basis vectors , , . ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle {\vec {e}}_{1}}$ ${\displaystyle {\vec {e}}_{2}}$ ${\displaystyle {\vec {e}}_{3}}$

properties

For the vector , the direction cosines are ${\displaystyle {\vec {v}}={\begin{pmatrix}v_{1}\\v_{2}\\v_{3}\end{pmatrix}}}$

${\displaystyle \cos \alpha _{1}={\frac {{\vec {v}}\cdot {\vec {e}}_{1}}{|{\vec {v}}|\,|{\vec {e}}_{1}|}}={\frac {v_{1}}{|{\vec {v}}|}}={\frac {v_{1}}{\sqrt {v_{1}^{2}+v_{2}^{2}+v_{3}^{2}}}}}$,

${\displaystyle \cos \alpha _{2}={\frac {{\vec {v}}\cdot {\vec {e}}_{2}}{|{\vec {v}}|\,|{\vec {e}}_{2}|}}={\frac {v_{2}}{|{\vec {v}}|}}={\frac {v_{2}}{\sqrt {v_{1}^{2}+v_{2}^{2}+v_{3}^{2}}}}}$,

${\displaystyle \cos \alpha _{3}={\frac {{\vec {v}}\cdot {\vec {e}}_{3}}{|{\vec {v}}|\,|{\vec {e}}_{3}|}}={\frac {v_{3}}{|{\vec {v}}|}}={\frac {v_{3}}{\sqrt {v_{1}^{2}+v_{2}^{2}+v_{3}^{2}}}}}$,

as can also be read from the colored triangles in the adjacent figure. Conversely, it can be expressed by its absolute value and the direction cosine, ${\displaystyle {\vec {v}}}$

${\displaystyle {\vec {v}}=|{\vec {v}}|{\begin{pmatrix}\cos \alpha _{1}\\\cos \alpha _{2}\\\cos \alpha _{3}\end{pmatrix}}}$.

If this is divided by, it turns out that the direction cosines are just the components of the unit vector in the direction of , ${\displaystyle |{\vec {v}}|}$ ${\displaystyle {\vec {e}}_{v}}$ ${\displaystyle {\vec {v}}}$

${\displaystyle {\vec {e}}_{v}={\frac {\vec {v}}{|{\vec {v}}|}}={\begin{pmatrix}\cos \alpha _{1}\\\cos \alpha _{2}\\\cos \alpha _{3}\end{pmatrix}}}$.

Because is ${\displaystyle |{\vec {e}}_{v}|=1}$

${\displaystyle \cos ^{2}\alpha _{1}+\cos ^{2}\alpha _{2}+\cos ^{2}\alpha _{3}=1}$.

Since the direction angles are limited to the range between and and the cosine is reversible in this interval, the three direction angles are also given with the direction cosines. ${\displaystyle 0}$ ${\displaystyle \pi }$

Individual evidence

1. ^ Gert Böhme : Introduction to higher mathematics (=  mathematics - lectures for engineering schools . Volume 2 ). Springer, 1964, p. 103–105 ( limited preview in Google Book search).
2. ↑ Eric W. Weisstein : Direction Cosine . In: MathWorld (English).