A man is walking in the north-east direction and wind appears to blow from north. If the man doubles his speed, wind appears at angle arccot 2 east of north. Find the actual direction of the wind.
The wind comes from the west. You can check that it works; anyway, here is how I found this out:
Let us denote by and the respective speeds of the wind and the man, in an orthonormal basis (E for east, N for north). Since the man goes NE, both components of are equal to some positive .
We know that the relative speed, i.e. the vector , is along the N direction, hence the E component is zero: .
Then we are told that the vector is (positively) colinear with . This is the content of the statement about "arccot 2" (draw a right triangle with perpendicular sides of length 1 and 2; the least angle measures ). As a consequence, . Replace by (as seen above), to get , hence .
Finally, (the same as in ), and , which means that the wind comes from the west.
I hope my explanations are clear.
Laurent.