SOLUTION: "While surveying a cave, a spelunker follows a passage 190 m straight west, then 210 m in a direction 45.0° east of south, and then 270 m at 30.0° east of north. After a fourth u

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Question 1161805: "While surveying a cave, a spelunker follows a passage 190 m straight west, then 210 m in a direction 45.0° east of south, and then 270 m at 30.0° east of north. After a fourth unmeasured displacement, she finds herself back where she started. Use vector components to find the magnitude and direction of the fourth displacement." (Direction is requested as degrees west of south)
I already found the magnitude of the resultant vector to be 127 m. However, my theta value was incorrect and the "west of south" is tripping me up. Could someone help?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
We start from point O%280%2C0%29 ,
moving 190 units West along the negative x-axis to point B
The components of vector OB are
OB%5Bx%5D=-190 and OB%5By%5D=0 .
We reach B%28-190%2C0%29
From there we go 210 units to point C in a direction 45%5Eo to the East of green%28S%29 .
That is 45%5Eo%2B270%5Eo=315%5Eo counterclockwise from the positive x-axis.
BC%5Bx%5D=210%2Acos%28315%5Eo%29=148.49 and BC%5By%5D=210%2Acos%28315%5Eo%29=-148.49 .
The coordinates of C are
x%5BC%5D=-190%2B148.49=-41.51 and y%5BC%5D=0%2B%28-148.49%29=-148.49
Then we head to point D for 270 units
in a direction that is 30%5Eo to the East of green%28N%29 .
That is 90%5Eo-30%5Eo=60%5Eo counterclockwise from the positive x-axis.
CD%5Bx%5D=270%2Acos%2860%5Eo%29=135 and CD%5By%5D=270%2Asin%2860%5Eo%29=233.83 .
The coordinates of C are
x%5BD%5D=-41.51%2B135=93.49 and y%5BD%5D=-148.49%2B233.83=85.33 .
Finally we head back to starting point O%280%2C0%29 .
The distance of that fourth displacement is
abs%28OD%29=sqrt%2893.49%5E2%2B85.33%5E2%29=highlight%28126.58%29
The angle alpha that OD makes with the positive x-axis is such that
tan%28alpha%29=95.33%2F93.49=0.9127 and alpha=42.39%5Eo
The angle theta the direction DO makes with the direction green%28S%29 is
90%5Eo-42.39%5Eo=highlight%2847.61%5Eo%29
and the direction DO is to the West of green%28S%29 .
It forms a counterclockwise angle of 270%5Eo-47.61%5Eo=222.39%5Eo with the positive x-axis
For verification:
The components of vector DO are
DO%5Bx%5D=0-93.5=-93.5=126.58%2Acos%28222.39%5Eo%29 and
DO%5By%5D=0-85.3=-85.3=126.58%2Asin%28222.39%5Eo%29