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Thread: Plane geometry problem

  1. #1
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    Plane geometry problem

    Find the equation of plane which contains the line $\displaystyle l:\left\{\begin{array}{l} x=t+2 \\y=2t-1\\z=3t+3 \end{array}\right.$, and makes the angle of $\displaystyle 2\pi/3$ with the plane $\displaystyle \pi:x+3y-z+8=0$.

    Actually don't have any idea so I need your help.
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Plane geometry problem

    One given you can already use:
    There's given the plane makes the angle $\displaystyle \frac{2\pi}{3}$ with the plane $\displaystyle \pi$, in general the angle $\displaystyle \theta$ between two planes is given by the formula:
    $\displaystyle \cos(\theta)=\frac{x_1\cdot x_2+y_1\cdot y_2+z_1\cdot z_2}{\sqrt{x_1^2+y_1^2+z_1^2}+\sqrt{x_2^2+y_2^2+z_ 2^2}}$

    Where $\displaystyle (x_1,y_1,z_1)$ and $\displaystyle (x_2,y_2,z_2)$ are the coordinates of the normal vectors of the two planes.
    You know the coordinates of the normal vector of the plane $\displaystyle \pi$ so you can use this given.

    About the line $\displaystyle l$ I have to think about, but the line is given in a parametric presentation so maybe I would try to reform it to a cartesian presentation by for example the substitution, let $\displaystyle y=t$. But I'm not sure about this.
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