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Math Help - Proving a triangle is right-angled.

  1. #1
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    Proving a triangle is right-angled.

    Show that the triangle with vertices P(1,2,2), Q(7,-1,-1) and R(3,0,8) is right-angled.

    My notes tell me that a triangle is only right angled if p.q = 0, but p.q = -11, p.r = 19, q.r = -29. I also tried to find out the individual angles and I got PQ = 120, PR = 42, QR= 118.

    Am I making an obvious mistake or just doing something stupidly wrong? Thanks for your help :]
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  2. #2
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    So

    P\left[\begin{array}{ccc}1\\2\\2\end{array}\right] Q\left[\begin{array}{ccc}7\\-1\\-1\end{array}\right] R\left[\begin{array}{ccc}3\\0\\8\end{array}\right]

    P.Q = (1.7) + (2.-1) + (2.-1) = 7 - 2 - 2 = 3
    P.R = (1.3) + (2.0) + (2.8) = 3 + 16 = 19
    Q.R = (7.3) + (-1.0) + (-1.8) = 21 - 8 = 12

    I believe the most likely situation is that the question or your information is wrong
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  3. #3
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    Thank you for your reply.

    The question is exactly as it is in the book, strange. Thanks for proving that it isn't right-angled :]
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  4. #4
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    Quote Originally Posted by scofield131 View Post
    Show that the triangle with vertices P(1,2,2), Q(7,-1,-1) and R(3,0,8) is right-angled.

    My notes tell me that a triangle is only right angled if p.q = 0, but p.q = -11, p.r = 19, q.r = -29. I also tried to find out the individual angles and I got PQ = 120, PR = 42, QR= 118.

    Am I making an obvious mistake or just doing something stupidly wrong? Thanks for your help :]
    1. The legs of the triangle are \overrightarrow{PQ}=(6,-3,-3) and \overrightarrow{PR}=(2,-2,6)

    2. \overrightarrow{PQ} \cdot \overrightarrow{PR} = 0

    3. Thus triangle PQR is a right triangle with the right angle at P.
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  5. #5
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    Ahhh, I forgot about that. Thank you so much :]
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