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Math Help - A triangle has vertices A (4,2,1), B (1,2,4), C (-1, 0, -4)

  1. #1
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    A triangle has vertices A (4,2,1), B (1,2,4), C (-1, 0, -4)

    Use dot products to find all three angles of the triangle.

    I already found the lengths of the three sides of the triangle: AB = 3sqrt(2), AC = 3sqrt(6) and BC = 6sqrt(2) and I already know the answers because I found them using trigonometry, but not sure how to do it using dot products.
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  2. #2
    MHF Contributor Unknown008's Avatar
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    To get the angles using dot product, you need the vectors, not the lengths.

    Find:

    the dot product of \vec{AB} and \vec{AC} to get the angle BAC.

    the dot product of \vec{BA} and \vec{BC} to get the angle ABC.

    the dot product of \vec{CB} and \vec{CA} to get the angle ACB.
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  3. #3
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    Could you please give the first one as an example? I already know AB is <-3, 0, 3> and AC is <-5,-2,-5> (that is AB and AC with the arrow on top)
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  4. #4
    MHF Contributor Unknown008's Avatar
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    You will need the formula:

    \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta

    Applying this, we get:

    \left(\begin{array}{c} -3 \\ 0 \\ 3\end{array}\right) \cdot \left(\begin{array}{c} -5 \\ -2 \\ -5\end{array}\right) = (3\sqrt2)(3\sqrt6) \cos\theta_1

    There theta1 is the angle A.

    Doing the dot product, we get:

    (-3)(-5) + (0)(-2) + (3)(-5) = 9\sqrt{12} \cos\theta_1

    \cos\theta_1 = \dfrac{15-15}{9\sqrt{12}}

    theta1 is 90 degrees.
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  5. #5
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    Thanks a lot, I really appreciate your help
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