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Math Help - Help with planes/lines of intersection

  1. #1
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    Help with planes/lines of intersection

    Can someone help with the following problem:

    Find an equation for the plane that passes through the point
    P(1,2,-2) and contains the line of intersection of the planes
    3x - y - z = 1 and x + y - z = 2. Show your work.
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  2. #2
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    Quote Originally Posted by messianic View Post
    Can someone help with the following problem:

    Find an equation for the plane that passes through the point
    P(1,2,-2) and contains the line of intersection of the planes
    3x - y - z = 1 and x + y - z = 2. Show your work.

    First find the intersection line of the two given planes: adding both equations you get x = \frac{1}{2}\,z+\frac{3}{4}, and substituting this in equation 2 you get y=\frac{1}{2}\,z+\frac{5}{4}, so the line, in parametric form, is \left\{\left(\frac{1}{2}\,t+\frac{3}{4}\,,\,\frac{  1}{2}\,t+\frac{5}{4}\,,\,t\right)\,,\,t\in\mathbb{  R}\right\} =\left(\frac{3}{4}\,,\,\frac{5}{4}\,,\,0\right)+t\  left(\frac{1}{2}\,,\,\frac{1}{2}\,,\,1\right)\,,\,  t\in\mathbb{R}.

    Now choose two different vectors in this line, say A=\left(\frac{3}{4}\,,\,\frac{5}{4}\,,\,0\right)\,  ,\,with\,\,\,t=0 , and B=\left(\frac{1}{4}\,,\,\frac{3}{4}\,,\,-2\right)\,,\,with\,\,\,t=-1, and then the plane you're looking for is P + r\overrightarrow{PA}+s\overrightarrow{PB}\,,\,r\,,  \,s\in\mathbb{R}.

    And this question belongs in pre-algebra-analytic geometry, not in calculus.

    Tonio
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