23. 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 1.30km 25.0º south of west; then 5.10 km straight east; then 1.70km 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?

R=7.34km ∠ 63.47° South of East.

Explanation:

We can either solve this graphically or by using the components of the given vectors. I'll solve it by using the components. So first we need to do a sketch of what the displacements will look like (See attached picture).

The piture is not drawn to scale, but it should help us visualize the directions better. I will name each displacement a different letter so we can distinguish them. We will say that if he goes north, he will have positive y-displacement, if he goes south, he will have negative y-displacement, if he goes east, he will have positive x-displacement and if he goes west, he will have negative x-displacement.

So having said this let's begin. So let's take the first displacement, displacement A:

2.50 km 45° north of west.  

He goes north here which means he has positive y-displacement and he goes west, which means he has negative x-displacement. Its components can be found by using the sin and cos functions, like this:

and we do the same with the other vectors.

B = 4.70km 60.0° south of east:

C = 1.30km 25.0° south of west:

D = 5.10km straight east:

E = 1.70km 5.00° east of north

In this case we need to flip the functions due to the direction of the displacement.

F = 7.20km 55.0° south of west:

G = 2.80km 10.0° north of east:

So now it's time to find the resulting vector, so we get:

So, now that we have the components of the resultant vector we can find magnitude of the displacement and the angle.

which yields

|R|=7.34 km

∠ =

∠ = -63.47°

So the final position relative to the island is:

7.34km 63.47° South of east

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