---

In an attempt to escape his island, Gilligan builds a raft and sets to sea. The wind shifts a great deal during the day, and he is blown along the following straight lines: 2.50 km 45.0º north of west; then 4.70 km 60.0º south of east; then 1.30 km 25.0º south of west; then 5.10 km straight east; then 1.70 km 5.00º east of north; then 7.20 km 55.0º south of west; and finally 2.80 km 10.0º north of east. What is his final position relative to the island?

---

Solution:

Gilligan’s displacement is characterized by 7 vectors. To determine his final position relative to the starting point, we simply need to add the vectors. To do this, we individually get the x and y components of each vector. This is presented in the table that follows:

Vector	X-Component	Y-Component
(1)	$-2.5\:\cos 45^{\circ} =-1.7678\:\text{km}$	$+2.5\:\sin 45^{\circ} =+1.7678\:\text{km}$
(2)	$+4.70\:\cos 60^{\circ} =+2.3500\:\text{km}$	$-4.70\:\sin 60^{\circ} =-4.0703\:\text{km}$
(3)	$-1.30\:\cos 25^{\circ} =-1.1782\:\text{km}$	$-1.30\:\sin 25^{\circ} =-0.5494\:\text{km}$
(4)	$+5.1000\:\text{km}$	$0$
(5)	$+1.70\:\sin 5^{\circ} =+0.1482\:\text{km}$	$+1.70\:\cos 5^{\circ} =+1.6935\:\text{km}$
(6)	$-7.20\:\cos 55^{\circ} =-4.1298\:\text{km}$	$-7.20\:\sin 55^{\circ} =-5.8979\:\text{km}$
(7)	$+2.80\:\cos 10^{\circ} =+2.7575\:\text{km}$	$+2.80\:\sin 10^{\circ} =+0.4862\:\text{km}$
Sum	$3.2799\:\text{km}$	$-6.5701\:\text{km}$

The table above indicates east and north as positive components, while west and south indicate negative components. The last row is the sum of the components. This is also the x and y components of the resultant vector.

To calculate the magnitude of the resultant, we simply use the Pythagorean Theorem as follows:

The direction of the resultant is calculated as follows:

Therefore, Gilligan is about 7.34 km directed 63.47° South of East from the starting island.

---