Complex Addition/Examples/Travel 1/Proof 1
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Example of Complex Addition
A man travels:
- $12$ kilometres northeast
then:
- $20$ kilometres $30 \degrees$ west of north
then:
- $18$ kilometres $60 \degrees$ south of west.
Assuming the curvature of the Earth to be negligible at this scale, at the end of this travel, he is $14.7$ kilometres in a direction $45 \degrees 49'$ west of north from his starting point.
Proof
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:
- $30 \degrees$ west of north is $30 + 90 \degrees$ north of east, and therefore an argument of $120 \degrees$
- $60 \degrees$ south of west is $60 + 180 \degrees$ north of east, and therefore an argument of $240 \degrees$.
Thus the problem reduces to finding the sum:
\(\ds P\) | \(=\) | \(\ds 12 \cis 45 \degrees + 20 \cis 120 \degrees + 18 \cis 240 \degrees\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {12 \cos 45 \degrees + 20 \cos 120 \degrees + 18 \cos 240 \degrees} + i \paren {12 \sin 45 \degrees + 20 \sin 120 \degrees + 18 \sin 240 \degrees}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {12 \frac {\sqrt 2} 2 + 20 \paren {-\dfrac 1 2} + 18 \paren {-\dfrac 1 2} } + i \paren {12 \sin 45 \degrees + 20 \sin 120 \degrees + 18 \sin 240 \degrees}\) | Cosine of $45 \degrees$, Cosine of $120 \degrees$, Cosine of $240 \degrees$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {12 \frac {\sqrt 2} 2 + 20 \paren {-\dfrac 1 2} + 18 \paren {-\dfrac 1 2} } + i \paren {12 \frac {\sqrt 2} 2 + 20 \frac {\sqrt 3} 2 + 18 \paren {-\frac {\sqrt 3} 2} }\) | Sine of $45 \degrees$, Sine of $120 \degrees$, Sine of $240 \degrees$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {6 \sqrt 2 - 19} + i \paren {6 \sqrt 2 + \sqrt 3}\) | after algebra |
Then:
\(\ds R \cis \theta\) | \(=\) | \(\ds \paren {6 \sqrt 2 - 19} + i \paren {6 \sqrt 2 + \sqrt 3}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds R\) | \(=\) | \(\ds \sqrt {\paren {6 \sqrt 2 - 19}^2 + \paren {6 \sqrt 2 + \sqrt 3}^2}\) | |||||||||||
\(\ds \) | \(\approx\) | \(\ds 14.7\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \theta\) | \(=\) | \(\ds \cos^{-1} \dfrac {6 \sqrt 2 - 19} {14.7}\) | |||||||||||
\(\ds \) | \(\approx\) | \(\ds \map {\cos^{-1} } {- 0 \cdotp 717}\) | ||||||||||||
\(\ds \) | \(\approx\) | \(\ds 135 \degrees 49'\) | ||||||||||||
\(\ds \) | \(\approx\) | \(\ds 45 \degrees 49' \text { west of north}\) |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Polar Form of Complex Numbers: $18 \ \text {(a)}$