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Math Help - Planar intersection

  1. #1
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    Planar intersection

    so I have a question I cannot answer in my calculus/vectors course, and it's driving me insane. if someone could just tell me how to find k i would be forever grateful.

    'Find the value of k such that the three planes will always intersect in a point, then find the point of intersection.'

    π1: x+2y-z-5=0
    π2: x+ky-3z+11=0
    π3 2x+y+z+10=0
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  2. #2
    A Plied Mathematician
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    From a linear algebra perspective, you get the intersection of a point if the system of equations you have there has exactly one solution, right? And when does that happen for the system

    A\mathbf{x}=\mathbf{b}, where

    A=\begin{bmatrix}1 &2 &-1\\ 1 &k &-3\\ 2 &1 &1\end{bmatrix}\qquad\text{and}\qquad\mathbf{b}=\b  egin{bmatrix}5\\-11\\-10\end{bmatrix}?
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  3. #3
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    Thats my problem, i'm drawing a complete blank. I've tried all the methods I know to narrow it down, and so far all I know is k cannot be 4
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  4. #4
    A Plied Mathematician
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    I wouldn't call that a complete blank! Especially since it's correct. k = 4 makes the matrix A singular, which means you'd get at least a line of intersection, not a single point.

    What's puzzling to me, and it sounds like to you, is that the problem asks for "the" value of k that makes the intersection just a point. That sounds to me like a typo. There are infinitely many values of k that make the intersection one unique point. Just with my own investigations here, solving the system by leaving k unspecified yields different solutions that depend on k.

    That's the best I can do for you, I'm afraid. I think it's a typo. They should have asked "What value of k makes the intersection a line?"
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