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The final position of Gilligan in kilometers and degrees south of east relative to the island is 9.07 km, 61.4° south of east.
A vector is a quantity or phenomenon that has two independent properties: magnitude and direction.
The term also denotes the mathematical or geometrical representation of such a quantity.
According to the question,
Suppose, the direction east as i, direction west as -i, direction north as j and direction south as j.
Therefore,
1. : 2.5 km 45° north of west
[tex]= 2.5 cos45 (-i) + 2.5 cos45 (j)\\ = -1.77(i) + 1.77(j)[/tex]
2. : 4.7 km 60° south of east
[tex]= 4.7 cos60 (i) + 4.7 sin60 (-j) \\= 2.35(i) - 4.07(j)[/tex]
3. : 5.1 km straight east
[tex]=5.1(i)[/tex]
4 : 7.2 km 55° south of west
[tex]= 7.2 cos55 (-i) + 7.2 sin55 (-j)\\ = -4.13(i) - 5.9(j)[/tex]
5: 2.8 km 5° north of east
[tex]= 2.8 cos5 (i) + 2.8 sin5 (j)\\= 2.79(i) + 0.24(j)[/tex]
The sum of all the above yields,
[tex]R = 4.34(i) + 7.96(-j)[/tex]
The magnitude of the resultant is:
[tex]|R| =\sqrt{4.34^{2} + (-7.96)^{2} }= 9.07[/tex]
The angle that the resultant makes is as follows.
θ= [tex]tan^-^1(\frac{7.96}{4.34} )\\[/tex]
θ = 61.4°
Hence,
Final position relative to the island is 9.07 km, 61.4° south of east.
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