Question

Can someone explain Question 3 from problem set 1, part 2 (the question about boat sails)? Thanks!

Answer 1

Problem 3, part a) asks you to show the "i" component (i.e. the x direction) points in the same direction as the wind vector (to the right)

part b) asks for the vector that represents the projection of w1 onto L_b
the unit length L_b has components \( < \cos( \alpha+\beta ), \ \sin( \alpha+\beta) >\)
we now need to scale that by the magnitude of w1 projected in that direction
namely \( | w1| \cdot \cos \beta\)
the magnitude of w1 is |w| cos \(\alpha\) = \( a \cos \alpha\)
thus we have
\[ a \cos \alpha \cos \beta< \cos( \alpha+\beta ), \ \sin( \alpha+\beta) >\]

they also ask what is required for the x component to be negative (point to the left)
we are told both \(\alpha\) and \( \beta\) are between 0 and 90 degrees, so their cosines will be positive. to get a negative number we need \( \cos( \alpha+\beta )<0\)
and that means \( \alpha + \beta > \frac{\pi}{2} \)