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Basic geometry - dot product/cart. lines

  1. Dec 16, 2012 #1
    1. The problem statement, all variables and given/known data
    We consider two points, B and I and a line 'a'.
    B(0,-4,-7) I(-2,-2,-5) and a: x = y+1 = (z-2)/2

    Determine the summits of A and C of triangle ABC knowing that:

    -Summit A belongs to the line 'a'
    -I is the foot of the height from A (perpendicular to BC)
    -The angle of A is equal to arccos(1/3)


    2. Relevant equations

    dot product

    3. The attempt at a solution

    I feel as if I'm really close, but keep getting the wrong answer. My process is as follows: The dot product of BI and AI is equal to zero. Knowing this, I can obtain an equation of the line AI and determine where it intersects with line 'a'. This is as far as I've gotten in my attempt to find A.

    Thanks a lot for any help you can provide!!
     
  2. jcsd
  3. Dec 16, 2012 #2

    I like Serena

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    Homework Helper

    Welcome to PF, boings!

    Can you set up a parametric equation for the line 'a'?
    That is, find a support vector and a direction vector?

    If you fill that in for A in your dot product, you'll get an equation from which you can find A....
     
  4. Dec 16, 2012 #3
    Hi and thanks!

    good point!

    (if k=constant)
    x= 0 + k
    y= -1 + k
    z= 2 + 2k

    Although, when I plug that in it doesn't make much sense. I proceed like this:

    AI (dot) BI = (-2, 2, 2)(dot)(-2, -2, -5)(0 + k, -1 + k, 2 + 2k)

    I think that this is already flawed in some sense
     
  5. Dec 16, 2012 #4

    I like Serena

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    Well, the vector AI is the difference of (-2, -2, -5) and (0 + k, -1 + k, 2 + 2k).
    So that is (-2 - k, -1- k, -7 - 2k).
    Then take the dot product and set it equal to zero...
     
  6. Dec 16, 2012 #5
    Alrighty, so I then get (4-2k, -2-2k, -14 - 4k)=0 which represents the line AI.

    This is where I get a little tripped up. Should I solve for k and replace into original equation of the line? The answer should be: A(-3, -4, -4)
     
  7. Dec 16, 2012 #6

    I like Serena

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    Hmm, (4-2k, -2-2k, -14 - 4k)=0 does not represent a line.
    How come you think that?

    The vector AI is represented by (-2 - k, -1- k, -7 - 2k) for some value of k.
    The vector BI is (-2, 2, 2).
    Their dot product has to be zero.

    The dot product is (-2-k) x -2 + (-1-k) x 2 + (-7-2k) x 2 = 0.
    Solve for k?
     
  8. Dec 17, 2012 #7
    Oh ok I get it! thank you so much.

    I see how that's not a line now, but rather the equation that relies the orthogonality between AI and BI. It sure looks like a a parametric equation of a line though ^^

    thanks again
     
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