The distance of the fourth leg is approximately 7.145 kilometers and the angle is approximately 58.879°.
Determinación de la magnitud y el ángulo de un vector a partir de una suma de vectores
El vector \(D\) es la suma de los tres vectores \(A\), \(B\) y \(C\), cuya operación se ve a continuación:
\(\vec D = -\vec A - \vec B - \vec C\) (1)
\(\vec D = -(6.4\cdot \cos 40^{\circ}, 6.4\cdot \sin 40^{\circ}) - (-5.10\cdot \cos 35^{\circ}, 5.10\cdot \sin 25^{\circ}) - (-4.80\cdot \cos 23^{\circ}, -4.80\cdot \sin 23^{\circ})\)
\(\vec D = (3.693, -6.117)\)
The magnitude is determined by Pythagorean theorem:
\(D = \sqrt{3.693^{2}+(-6.117)^{2}}\)
\(D \approx 7.145\,km\)
The direction is found by this inverse trigonometric relationship:
\(\theta_{D} = \tan^{-1} \frac{6.117\,km}{3.693\,km}\)
\(\theta_{D} \approx 58.879^{\circ}\)
The distance of the fourth leg is approximately 7.145 kilometers and the angle is approximately 58.879°.
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