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

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

OCW Scholar - Multivariable Calculus
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  • phi
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 \( \alpha + \beta > \frac{\pi}{2} \)

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