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Mathematics-Online course: Prepcourse Mathematics - Linear Algebra and Geometry - Lines and Planes

Air Lanes

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Simplifying let us assume that aircrafts fly directly from the starting point to the destination and that the flight path consists in line segments. Then the ascending flight path of a flight from Stuttgart $ S$ to Kopenhagen is given by

$\displaystyle g: \overrightarrow{SX} = s\left(\begin{array}{r}8\\ 16\\ 1\end{array}\right)\,,\quad s\in [0,8] $

and the ascending flight path of a flight from Frankfurt ($ F$ ) to Cairo is given by

$\displaystyle h: \overrightarrow{FX} = t\left(\begin{array}{r}34\\ -31\\ 4\end{array}\right)\,, \quad t\in[0,3]\,. $

If we attach the origin of our coordinate system to Stuttgart, then Frankfurt has the coordinates $ F=(-50,150,-1/4)$ .

Hence, the distance between the two flight paths is


$\displaystyle d$ $\displaystyle =$ $\displaystyle \frac{ \left\vert\left(\left(\begin{array}{r}-50\\ 150\\ -1/4\end... ...}\right)\times \left(\begin{array}{r}34\\ -31\\ 4\end{array}\right)\right\vert}$  
  $\displaystyle =$ $\displaystyle \frac{ \left\vert\left(\begin{array}{r}-50\\ 150\\ -1/4\end{array... ...\right\vert} {\sqrt{95^2+2^2+792^2}} =\frac{4252}{\sqrt{636293}}\approx 5.33\,.$  

\includegraphics[width=10.4cm]{b_flugkorridore_bild_en}

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  automatically generated 9/18/2007