SOLUTION: I have a problem the question is find the magnitude of the sum, u+v, to the nearest tenth and give the direction by specifying to the nearest degree the angle that it makes the vec

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Question 818134: I have a problem the question is find the magnitude of the sum, u+v, to the nearest tenth and give the direction by specifying to the nearest degree the angle that it makes the vector u.
|u| = 12, |v| = 15, theta = 120 degrees
the final answer im suppose to come up with is 13.7 for the magnitude and the direction is 71 degrees.
but my work i got 19.2 for the magnitude and the direction is 39 degrees where did I go wrong?
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
Here's one way to solve this...
  1. On a graph, label the origin as "O".
  2. Plot the point (15, 0) and label it V. This makes OV an image of vector v.
  3. In component form, v = 15i + 0j.
  4. Draw a segment from O at an angle of 120 degrees from OV. This segment should extend into the second quadrant. Label the end of this segment as U and label its length as 12. This make OU an image of vector u.
  5. Draw a vertical segment down from U to the negative part of the x-axis. Label this point A.
  6. Draw a segment from A to O. This completes the triangle OUA. And because UA is vertical and OA is horizontal, UA and OA form a right angle. This makes triangle OUA a right triangle.
  7. Since angle UOV is 120 degrees, angle UOA is 60 degrees (because these two angles form a linear pair which are always supplementary).
  8. Since angle UOA is 60 degrees and triangle UOA is a right triangle, triangle UOA is a 30/60/90 right triangle.
  9. Since triangle UOA is a 30/60/90 right triangle we can use the pattern for this type of triangle to find the remaining sides. With a hypotenuse of 12, the side opposite the 30 degree angle, OA, will be half as much: 6. Label the length of OA as 6.
  10. And the side opposite 60 degrees is times the side opposite 30: or . Label the length of UA as
  11. Using UA and OA we can now express vector u in component form:
    (Note: The i component is negative because it goes to the left from the origin.)
  12. We can now add vectors u and v using the components:
  13. We can now find the magnitude of u+v:
    which is approximately 13.7477
  14. We can also now find the direction:
    which is approximately 49.1066 degrees.
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