Complex Addition/Examples/Travel 2
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Example of Complex Addition
An airplane travels:
- $150$ kilometres southeast
then:
- $100$ kilometres due west
then:
- $225$ kilometres $30 \degrees$ north of east
then:
- $323$ kilometres northeast.
Assuming the curvature of Earth to be negligible at this scale, at the end of this travel, the plane is $490$ kilometres in a direction $28.7 \degrees$ north of east from its starting point.
Proof 1
Let the route of the traveller be embedded in the complex plane.
Let $P$ be the final location of the traveller.
The given directions can be translated into absolute arguments thus:
Thus the problem reduces to finding the sum:
\(\ds P\) | \(=\) | \(\ds 150 \cis 315 \degrees + 100 \cis 180 \degrees + 225 \cis 30 \degrees + 323 \cis 45 \degrees\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {150 \cos 315 \degrees + 100 \cos 180 \degrees + 225 \cos 30 \degrees + 323 \cos 45 \degrees} + i \paren {150 \sin 315 \degrees + 100 \sin 180 \degrees + 225 \sin 30 \degrees + 323 \sin 45 \degrees}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {150 \frac {\sqrt 2} 2 + 100 \paren {-1} + 225 \paren {\dfrac {\sqrt 3} 2} + 323 \paren {\dfrac {\sqrt 2} 2} } + i \paren {150 \sin 315 \degrees + 100 \sin 180 \degrees + 225 \sin 30 \degrees + 323 \sin 45 \degrees}\) | Cosine of $315 \degrees$, Cosine of $30 \degrees$, Cosine of $45 \degrees$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {150 \frac {\sqrt 2} 2 + 100 \paren {-1} + 225 \paren {\dfrac {\sqrt 3} 2} + 323 \paren {\dfrac {\sqrt 2} 2} } + i \paren {150 \frac {-\sqrt 2} 2 + 100 \paren {0} + 225 \paren {\dfrac 1 2} + 323 \paren {\dfrac {\sqrt 2} 2} }\) | Sine of $315 \degrees$, Sine of $30 \degrees$, Sine of $45 \degrees$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {106.07 - 100 + 194.86 + 228.40} + i \paren {-106.07 + 0 + 112.50 + 228.40}\) | after algebra and calculation | |||||||||||
\(\ds \) | \(=\) | \(\ds 429.33 + i 234.83\) | after calculation |
Then:
\(\ds R \cis \theta\) | \(=\) | \(\ds 429.33 + i 234.83\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds R\) | \(=\) | \(\ds \sqrt {429.33^2 + 234.83^2}\) | |||||||||||
\(\ds \) | \(\approx\) | \(\ds 489.35\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \theta\) | \(=\) | \(\ds \tan^{-1} \dfrac {234.83} {429.33}\) | |||||||||||
\(\ds \) | \(\approx\) | \(\ds \map {\cos^{-1} } {0 \cdotp 547}\) | ||||||||||||
\(\ds \) | \(\approx\) | \(\ds 28.7 \degrees\) |
$\blacksquare$
Proof 2
By plotting the points in a graphics package, or on paper with a ruler and protractor:
$\blacksquare$