Expert: Sherry Wallin - 5/27/2010

Question
Find the magnitude of w to three significant digits and its bearing to the nearest tenth of a degree.
#1)
u:magnitude 23.0, bearing 215 degrees
v:magnitude 14.5, bearing 105 degrees
w:u+v

#2)
u:magnitude 621, bearing 305 degrees
v:magnitude 336, bearing 15 degrees
Find w if 3(w+u)=2(w-v)

#3)
u:magnitude 218, bearing 22 degrees
v:magnitude 170, bearing 112 degrees
w:u-v

#4)
u:magnitude 1850, bearing 125 degrees
v:magnitude 2960, bearing 25 degrees
w:u+2v


Thank you so much!

Answer
Allison, I will explain how to do a couple of these problems but it would be helpful if you showed me how you were approaching these problems so I could help you learn how to do them.

#1 w = u + v means that you need to rewrite u and v in terms of their components, i.e. in terms of the cosines and sines. a magnitude of 23 means you will need to multiply the new representation by 23 but you need to pay attention to the angle and the sign of the cosine and sine function where that angle lies. 215 degrees is in the 3rd quadrant and both the cosine and the sine function are negative so this is just like multiplying the new representation by minus the magnitude or -23: Also there is an angle that is corresponding to 215 degrees and that is 215-180 = 35 degress so this is how you would rewrite u: -23(cos(35)+sine(35)) and then for v you need to be looking at the 2nd quadrant because that is where 105 is and we know that the cosine is negative there and the sine is positive there so it becomes (remember 180-105 = 75 deg):
-14.5*cos(75)+14.5*sin(75). Now you can add u+v
= -23(cos(35)+sin(35))+-14.5*cos(75)+14.5*sin(75)
= -23*cos(35)+-14.5*cos(75)+-23*sin(35)+14.5*sin(75)   we need to add like components
=-23(.819) +-14.5(.259) (horizontal component your cosines) -23(.574)+14.5(.966) (vertical sines)
=-22.593 this is the 'new' x component of the sum and .805 is the 'new' y component
Now use pythagoreans theorem to determine the new vectors magnitude:
(-22.593)^2 + (.805)^2 = magnitude^2 -> 510.444 + .648 = 511.092 = magnitude^2 so take it's square root: sqrt(511.092)~= 22.607 this is the new vectors magnitude

To find the new bearing we do the following: first determine where the new vector lies (which quadrant). You can either draw the two vectors you addedtail to head and then see where this new bearing is or just use the fact that the cosine of the new vector is negative and the sine of the new vector is positive and realize that the cosine represents the x value and the sine represents the y value so you know x is neg and yi pos in the 2nd quadrant. Now the formula for finding the angle is: arctangent(ysum/xsum) = arctan(.805/-22.593) ~= -63.89 deg which is in the 4th quadrant. We want the 2nd quadrant so we add 180 degees and find the correct angle for the resultant vector is 116 deg (rounded to the nearest degree). So the new vector has a magnitude of 22.607 and a bearing of 116 degrees.

#2: The problem tells you that 3(w+u) = 2(w-v) solve this algebraically
3w + 3u = 2w - 2v -> w = -3u -2v
The rest of your problems can be done the same except is you have 3u first multiple the magnitude of u by 3 because the magnitude represent the length of the vector so 3u = u + u + u.

Math Prof