The direction of motion of the person relative to the water is 16.7° north of east.
Why?
We can solve the problem by applying the Pitagorean Theorem, where the first speed (to the north) and the second speed (to the east) corresponds to two legs of the right triangle formed with them. (north and east directions are perpendicular each other)
We can calculate the angle that give the direction using the following formula:
[tex]Tan(\alpha)=\frac{NorthSpeed}{EastSpeed}\\\\Tan(\alpha)^{-1}=Tan(\frac{NorthSpeed}{EastSpeed})^{-1}\\\\\alpha=Tan(\frac{NorthSpeed}{EastSpeed})^{-1}[/tex]
Now, substituting the given information we have:
[tex]alpha=Tan(\frac{NorthSpeed}{EastSpeed})^{-1}[/tex]
[tex]\alpha =Tan(\frac{3.6\frac{m}{s} }{12\frac{m}{s} })^{-1}\\\\\alpha =Tan(0.3)^{-1}=16.69\°(North-East)=16.7\°(North-East)[/tex]
Hence, we have that the direction of motion of the person relative to the water is 16.7° north of east.
Have a nice day!
