The route followed by a hiker consists of three displacement vectors a, b, and c. Vector a is along a measured trail and is 1550 m in a direction 29.0° north of east. Vector b is not along a measured trail, but the hiker uses a compass and knows that the direction is 41.0° east of south. Similarly, the direction of vector c is 14.0° north of west. The hiker ends up back where she started, so the resultant displacement is zero, or a+b+c= 0. Find the magnitudes of vector b and c.


I am able to set up the problem pretty easily using the component method and I end up with these two equations 1550cos(29)+Bsin(41)-cos(14)=0 & 1550sin(29)-Bcos(41)+Csin(14)=0. I'm not really sure how to solve these two equations. I think that you set both of them equal to 0 but that really didn't help me. Also I'm not sure how you would use trig and the pathogrean theorem when there are three vectors. Please can you help me?

Your technique seems valid- you ended up with 2 equations in 2 unknown (B and C), so you should be able to solve for B and C. There is a typo in your first formula -you're missing the factor C in front of the cos(14).

Here's another approach: since you know all the bearings involved you can figure out the interior angles of the triangle that the hiker followed (for example, the angle between A and C is 29+14 = 43 degrees), and since you know the length of A, you can use the law of sins to calculate lengths B and C.