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Math Help - Algebra - Vectors

  1. #1
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    Smile Algebra - Vectors

    a) By looking at their intersection and their direction vectors, show that the following lines are skew:
    (x,y,z) = (2, -1, 6) + t(1, 2, 0)
    and
    (x, y, z) = (1, 4, 4) + u(2, -3, 1).


    b) Find the closest point to the origin on the line (x,y,z) = (2, -1, 6) + t(1, 2, 0) and the shortest distance from the origin to that line.
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  2. #2
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    For b), first rewrite the line as \displaystyle (x, y, z) = (2 + t, -1 + 2t, 6).

    The distance from the origin to the line is \displaystyle D = \sqrt{(2 + t - 0)^2 + (-1 + 2t - 0)^2 + (6-0)^2}

    \displaystyle = \sqrt{(2 + t)^2 + (-1 + 2t)^2 + 6^2}

    \displaystyle = \sqrt{4 + 4t + t^2 + 1 - 4t + 4t^2 + 36}

    \displaystyle = \sqrt{5t^2 + 41}.


    The minimum distance will the value of \displaystyle t for which the derivative is \displaystyle 0 and where the second derivative is positive.
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  3. #3
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    What ideas have you had for a)?
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