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Math Help - Vector(3-Space) Word Problem Help

  1. #1
    Junior Member Morphayne's Avatar
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    Vector(3-Space) Word Problem Help

    Problem:

    Determine if the points P(3, -2, -7), Q(0 , 4, 2), R(-1, 3, -1) and S(5, -1, -3) are coplanar.

    Answer in text: Yes

    Additional Comments:

    Please help me, I have a test tomorrow that basically decides my future.
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  2. #2
    MHF Contributor Reckoner's Avatar
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    Quote Originally Posted by Morphayne View Post
    Problem:

    Determine if the points P(3, -2, -7), Q(0 , 4, 2), R(-1, 3, -1) and S(5, -1, -3) are coplanar.
    One very fast and easy way to check if a set of four points are coplanar is by putting them in a matrix and checking the determinant.

    Four points, \left(x_1, y_1, z_1\right),\;\left(x_2, y_2, z_2\right),\;\left(x_3, y_3, z_3\right),\text{ and }\left(x_4, y_4, z_4\right) are coplanar if and only if

    \left|\begin{matrix}<br />
x_1 & y_1 & z_1 & 1\\<br />
x_2 & y_2 & z_2 & 1\\<br />
x_3 & y_3 & z_3 & 1\\<br />
x_4 & y_4 & z_4 & 1\\<br />
\end{matrix}\right| = 0

    I don't have a proof of this off-hand, but it should work.
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  3. #3
    Moo
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    A Cute Angle Moo's Avatar
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    Jhevon is going to kill me for that

    ---------------------------

    By the way ^^

    I suppose you're in high school and because you talked about linear systems, we'll do it this way :

    The equation of a plane is ax+by+cz+d=0

    So write the linear system of equations assuming that P, Q, R and S are in this plane. Then solve for a,b,c and d.
    If it has a (unique) solution, it will mean that the points are coplanar I'll let you do the working, because I really have to go to bed


    Actually, this is equivalent to Reckoner's method, but his is for higher grades
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  4. #4
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    Here is an easy if tedious way to check.
    If \overrightarrow {PS}  \cdot \left( {\overrightarrow {PQ}  \times \overrightarrow {PR} } \right) = 0 then the four points are coplanar.
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