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Determine the acute angle between two intersecting lines

  1. Aug 26, 2011 #1
    1. The problem statement, all variables and given/known data
    A basketball gymnasium is 37 meters high, 80 meters wide and 200 meters long. For a half time stunt, the cheerleaders want to run two strings, one from each of the two corners above one basket to the diagonally opposite corners of the gym floor. What is the cosine of the acute angle made by the strings as they cross? Round your answer to 3 decimal places.


    2. Relevant equations
    I believe that you just have to equate:
    [itex]\left\|\vec{v}\right\| \left\|\vec{w}\right\|\cos{\theta} = \vec{v} \cdot \vec{w}[/itex]

    I thought that the sign of the dot product determines whether an angle is obtuse or acute.


    3. The attempt at a solution

    Vectors:
    [itex]\vec{v} = \left\{200\hat{i}, 80\hat{j}, -37\hat{k}\right\}[/itex]
    [itex]\vec{w} = \left\{200\hat{i}, -80\hat{j}, -37\hat{k}\right\}[/itex]

    Magnitudes:
    [itex]\left\|\vec{v}\right\| = \sqrt{200^2 + 80^2 + (-37)^2} = \sqrt{47769}[/itex]
    [itex]\left\|\vec{w}\right\| = \sqrt{200^2 + (-80)^2 + (-37)^2} = \sqrt{47769}[/itex]
    [itex]\left\|\vec{v}\right\| \left|\vec{w}\right\| = 47769[/itex]

    Dot Product:
    [itex]\vec{v} \cdot \vec{w} = 200*200 + (-80)*80 + (-37)*(-37) = 34969[/itex]

    Equating the sides:
    [itex]\cos{\theta} = \displaystyle\frac{\vec{v} \cdot \vec{w}}{\left\|\vec{v}\right\| \left\|\vec{w}\right\|}[/itex]

    [itex]\cos{\theta} = \displaystyle\frac{34969}{47769}[/itex]

    After solving for theta by taking the arccos, I have 0.749. With that all said and done, it is not the correct answer. I am not entirely sure where I went wrong. I checked my math and it seems to point to being the correct answer.
     
  2. jcsd
  3. Aug 26, 2011 #2

    tiny-tim

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    welcome to pf!

    hi justinvh! welcome to pf! :smile:

    (have a theta: θ :wink:)
    erm :redface:

    you're asked for cosθ, not θ :wink:
     
  4. Aug 26, 2011 #3
    Re: welcome to pf!

    Oh my dear lord. I can't believe I missed that. I've been driving myself crazy for the last day and a half. Thank you for pointing that out.
     
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