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Math Help - Vector Equations

  1. #1
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    Vector Equations

    2 questions

    1. Determine the value of a for which the following planes intersect in a line:

    x-2y-z=0
    x+9y-5z=0
    ax-y+z=0

    2. Given the Scaler Equation of a plane, determine a corresponding vectot equation. 2x+5y-z+10=0

    Lastly any good school books on vectors?
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  2. #2
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    Quote Originally Posted by someone21 View Post
    2 questions

    1. Determine the value of a for which the following planes intersect in a line:

    x-2y-z=0
    x+9y-5z=0
    ax-y+z=0

    2. Given the Scaler Equation of a plane, determine a corresponding vectot equation. 2x+5y-z+10=0

    Lastly any good school books on vectors?
    to #1:

    If
    p_1: x-2y-z=0 then the normal vector of p_1 is \overrightarrow{n_1} = (1, -2, -1)

    p_2: x+9y-5z=0 then the normal vector of p_2 is \overrightarrow{n_2} = (1, 9, -5)

    p_3: ax-y+z=0 then the normal vector of p_3 is \overrightarrow{n_3} = (a, -1, 1)

    The direction vector of the intersection line of the first two planes is:

    \overrightarrow{n_1} \times \overrightarrow{n_2}=(19, 4, 11)

    The normal vector \overrightarrow{n_3} must be perpendicular to this direction vector:

    \overrightarrow{n_3} \cdot (19, 4, 11) = 0~\implies~ (a, -1, 1) \cdot (19, 4, 11) = 0

    which will yield a = -\frac7{19}

    to #2:

    Given
    2x+5y-z+10=0
    That means:
    z = 2x+5y+10

    Now substitute x = r and y = s

    \begin{array}{l}x = r \\ y = s \\ z = 10 +2r+5s\end{array} ...... \implies ...... (x, y, z) = (0, 0, 10) + r(1, 0 , 2) + s( 0, 1, 5)
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