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Math Help - Vector problem involving two planes and point of origin!!!

  1. #1
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    Vector problem involving two planes and point of origin!!!

    hello, I am studying for my exams and I came across a question that is really giving me trouble:

    The planes: 2x + 3y + z = 2 and 5x - 2y + 2z = -4 are given. Find the scalar equation of the plane that contains the line of intersection of plane 1 and 2 and that passes through the origin.

    I tried to use Ax + By + Cz + D + k(A2x + B2y + C2z + d2) = 0 but it didn't really seem to work out.

    for quick reference:

    scalar equation = ax + by + cz + d = 0

    the answer at the back of the book is 9x + 4y + 4z = 0 but it might be wrong.

    I will really appreciate some help.
    Thanks a lot!!!
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  2. #2
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    Pick a point in the intersection: \left( {0,1, - 1} \right).
    This cross product \left( {\left\langle {2,3,1} \right\rangle  \times \left\langle {5, - 2,2} \right\rangle } \right) \times \left\langle {0,1, - 1} \right\rangle will give you the normal.

    BTW the answer in the textbook is correct.
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  3. #3
    Junior Member Morphayne's Avatar
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    Quote Originally Posted by Plato View Post
    Pick a point in the intersection: \left( {0,1, - 1} \right)...
    How did you find that point?
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  4. #4
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    Quote Originally Posted by Morphayne View Post
    How did you find that point?
    The line of intersection is found by solving the equations 2x + 3y + z = 2 and 5x - 2y + 2z = -4 simultaneously.

    But you only need a point on this line. In other words, you only need one concrete solution from the infinite number of solutions available. So .....

    Choose one of the variables, x say. Pick a convenient value for x, x = 0, say. Substitute x = 0 into the equations for the two planes. Solve the resulting equations simultaneously .....
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  5. #5
    Junior Member Morphayne's Avatar
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    Quote Originally Posted by mr fantastic View Post
    Solve the resulting equations simultaneously .....
    What does that mean? Use substitution/elimination?
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  6. #6
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    Quote Originally Posted by Morphayne View Post
    What does that mean? Use substitution/elimination?
    Use whatever method you prefer.
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