Question

A bird flies from its nest 10 km in the direction 30◦ north of east, where it stops to rest on a tree. Then it flies 6 km in the direction due southeast and lands on a pole. If you assume that the nest is located at the origin, find the location of the pole.

Answer 1

Compute the resultant vector:

$(10\,\mathrm{km})(\cos(30^\circ)\,\vec\imath + \sin(30^\circ)\,\vec\jmath) + (6\,\mathrm{km}) (\cos(-45^\circ)\,\vec\imath + \sin(-45^\circ)\,\vec\jmath) \\\\ ~~~~ = \bigg((3\sqrt2+5\sqrt3)\,\vec\imath + (5 - 3\sqrt2) \,\vec\jmath\bigg) \, \mathrm{km} \approx \boxed{\bigg(12.9029\,\vec\imath + 0.7574\,\vec\jmath\bigg) \,\mathrm{km}}$

That is, the pole is located approximately 12.9 km east and 0.76 km north of the origin.