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Old Nov 13th 2012, 06:57 PM   #1
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Adding vectors by components

Hello, I don't have a question about how to solve this problem, but I can not for the life of me figure out how the author comes up with θ. Here is the problem:

In a road rally, you are given the following instructions: from the starting point use available roads to drive 36 km due east to checkpoint, then 42 km due north, and then 25 km northwest.

Q) What are the magnitude and orientation of your displacement "d" from the starting point?

They also provide a figure that includeds the angle (below), but I can't figure out how it's derived.
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Old Nov 14th 2012, 04:52 AM   #2
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To add vectors break each portion of the legs of the journey into x-axis (east-west) and y-axis (north-south) components. So the first leg starts at point O and goes 36 Km east, or +36 in the x direction to point A. The second leg is +42 in the y direction to point B. The third leg is 25 in the northwest direction to point C , which is a bearing of 135 degrees from B, so the x-component of that leg is 25 times cosine 135 degrees, or -25(sqrt2)/2. The y component is 25sin(135), or 25(sqrt2)/2. Now add the x components of all three vectors and you get X = 36-25(sqrt2)/2 = 18.32. The sum of all the y components is 42+25(sqrt2)/2 = 59.68. Hence the coordinates of point C in the figure is (18.32,59.68).

The magnitude of the vector from O to C is found using the Pythagorian Theorem: d=sqrt(X^2+Y^2) = sqrt(18.32^2+59.68^2)=62.4. And the angle theta is that arctangent of Y/X, or Arctan(59.68/18.32) = 72.9 degrees.

Let me know if there is any portion of this that is confusing.
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Old Nov 14th 2012, 09:25 AM   #3
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So, the angle of 135 is used to calculate the components of vector C because this is the angle vector C makes with increasing x, correct?

Did you calculate 135, or did you just use that because it's listed in the figure?
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Old Nov 14th 2012, 10:04 AM   #4
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Yes, 135 degrees is the angle between the third leg and the positive x-direction. By using angles based on the x-axis like this you can apply cosine of the angle to find the horizontal component and sine of the angle to find the vertical component. How did I know it's 135? Because we were told that the this leg is in the northwest direction, and 135 degrees is halfway between north (which is 90 degrees) and west (which is 180 degrees).
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Old Nov 14th 2012, 10:17 AM   #5
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Aaaahh it's always the obvious parts that throw me. I Wasn't thinking the implied literal NW.

Thanks for your help. May the ma be with you!
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