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Math Help - Determinant & vector geometry problem

  1. #1
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    Determinant & vector geometry problem

    1.) Show that

    det ( 1+x 2 3 4
    1 2+x 3 4
    1 2 3+x 4
    1 2 3 4+x)

    = x^3 (x+10)

    It should be easy but I dont know why I cannot get the answer
    Please help!

    2.) Consider the triangle ABC formed by the points A(3,2,1), B(4,4,2) and C(6,1,0). Find the coordinates of point D on BC such that AD is perpendicular to BC. (ans: 1/17 (80 50 22)T)

    THANKS A LOT!!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by coeyz View Post
    1.) Show that

    det ( 1+x 2 3 4
    1 2+x 3 4
    1 2 3+x 4
    1 2 3 4+x)

    = x^3 (x+10)

    It should be easy but I dont know why I cannot get the answer
    Please help!
    What you tried some elimination on the matrix?
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  3. #3
    MHF Contributor

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    2.) Consider the triangle ABC formed by the points A(3,2,1), B(4,4,2) and C(6,1,0). Find the coordinates of point D on BC such that AD is perpendicular to BC. (ans: 1/17 (80 50 22)T)
    Any point on BC has coordinates x= 4+ (6- 4)t= 4+ 2t, y= 4+ (1- 4)t= 4- 3t, and z= 2+ (0- 2)t= 2- 2t.

    Let D have coordinates (x, y, z). Then the vector AD is (x- 3)i+ (y- 2)j+ (z- 1)k and BC is 2i- 3j- 2k. The dot product of those two vectors is 0. Replacing x, y, and z with there expression in t gives a single equation for t.
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