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Vector problem: Find region shaded by fence

  1. Nov 1, 2009 #1
    1. The problem statement, all variables and given/known data
    A vertical fence stands on the ground, and the sun is shining on it.
    The ground is the xy-plane, and the top of the fence is the line through the point (0,0,1) in the
    direction of < 3, 1, 0 >. The sun’s rays are pointing in the direction of the vector < 2,−1,−3 >.
    Find the region on the ground that is shaded by the fence.

    2. Relevant equations
    A dot B = |A||B|cosθ
    A x B = |A||B|sinθ

    3. The attempt at a solution
    I'm not sure how to visualize the problem.
    Also, what is meant by "find the region"? Are they asking for the area of the shaded ground?
    Can someone shed some light on this problem, or help me set it up?
     
  2. jcsd
  3. Nov 1, 2009 #2

    lanedance

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    so the ground in the xy plane, and the fence is one unit high

    think of the right triangle formed by the fence (vertical), the ground (horizontal) with the hypotenuse direction, given by the direction of the sunlight

    for the region, you could think of it bounded by 2 lines, one the base of the fence, the other the edge of the shadow
     
  4. Nov 1, 2009 #3
    But what exactly is it that I'm finding? I can't find the area since I don't know the length of the fence. So am I just finding the distance of the shadow from the edge of the fence?

    My work: I started the vector <2, -1, -3> at the point (0, 0, 3).
    I then used the pythagorean theorem: c = sqrt[2^2 + (-1)^2] = sqrt5
    I then found an equation: y = (-3/sqrt5)x + 3
    I moved the fence up so that the fence is a line through (0, 0, 3) and the bottom of the fence is the line through (0, 0, 2)
    I set the equation equal to 2, and solved for x, and got sqrt5 / 3
     
    Last edited: Nov 1, 2009
  5. Nov 1, 2009 #4

    lanedance

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    where did you get (0,0,3)?

    the top of the fence is given by the line through (0,0,1) along direction (3,1,0). The ground is the xy plane.

    This means the base of the fence is is given by the line through (0,0,0) along direction (3,1,0).

    Find the line representing the edge of the shadow. The shaded region will be bounded by the base of the fence on one side, and the edge of the shadow on the other, so by 2 parallel lines. Unless the length of teh fence is given, I would probably assume it is infinite
     
  6. Nov 1, 2009 #5
    I moved all the vectors up so that it would be easier to work with, but all the directions of the vectors are the same.
     
  7. Nov 1, 2009 #6

    lanedance

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    why are they easier to deal with moved up?

    If you know the line of the bottom of the fence, all you need is to find a vector to shift it, to find the shadow edge.

    start with fence f = (0,0,1)

    the sun direction is s = (2,-1,-3)/3 = (2/3,-1/3,-1), if we divide it by three it gives the same vertical relief as the fence (ie. when the shadow hits the ground).

    the vector represting the shadow on the ground is the sum of these
    b = f + s = (0,0,1) + (2/3,-1/3,-1) = (2/3,-1/3,0)

    which has the same length as you found
    |b| = (2/3,-1/3,0) = sqrt(5)/3
     
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