To find the resultant vector you need to add the horizontal and vertical components of each individual vector.
Cosine for horizontal, Sine for vertical.
[tex]v = \ \textless \ |v|cos \theta, |v| sin \theta\ \textgreater \ [/tex]
Where |v| is magnitude or length of vector.
Lets look at first ranger station. First vector has magnitude 4.5 with angle of 90.
Next has magnitude of 8.1 with angle of 125.
[tex]F_1 = \ \textless \ 4.5 cos(90)+8.1 cos(125) , 4.5 sin(90) + 8.1 sin(125) \
\textgreater \ [/tex]
[tex]F_1 = \ \textless \ -4.646, 11.135\ \textgreater \ [/tex]
Find magnitude:
[tex]|F_1| = \sqrt{((-4.646)^2+(11.135)^2} = 12.065[/tex]
Do the same with 2nd ranger station:
[tex]F_2 = \ \textless \ 7.5 cos(20) + 5.3 cos(100), 7.5 sin(20) + 5.3 sin(100)\ \textgreater \ [/tex]
[tex]F_2 = \ \textless \ 6.127, 7.784\ \textgreater \ [/tex]
[tex]|F_2| = \sqrt{(6.127)^2 + (7.784)^2} = 9.9[/tex]
Therefore the 2nd ranger station is closest to starting point.
