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Math Help - line of intersection of planes

  1. #1
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    line of intersection of planes

    Hi..how do I find the unit vector parallel to the line of intersection of planes, 2x+2y-z=6 and x+2y+z=2. So far I got 2x+2y-z-6+\lambda(x+2y+z-2)=0..
    THen i need to find the parametric form of the equation of a line passing thru the point A[2,2,1] that is parallel to the line of the intersection of the 2 planes above.

    THanks
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  2. #2
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    Quote Originally Posted by ashes View Post
    Hi..how do I find the unit vector parallel to the line of intersection of planes, 2x+2y-z=6 and x+2y+z=2. So far I got 2x+2y-z-6+\lambda(x+2y+z-2)=0..
    THen i need to find the parametric form of the equation of a line passing thru the point A[2,2,1] that is parallel to the line of the intersection of the 2 planes above.

    THanks
    The normal vectors of the planes are perpendicular to the direction of the line of intersection. Thus:

    \vec d = (2, 2, -1) \times (1, 2, 1) = (4, -3, 2)

    Then the equation of a line passing through A in the direction of \vec d is:

    \vec r = (2, 2, 1) + s \cdot (4, -3, 2) ...... which will yield:

    l:\left\{ \begin{array}{lcr}x&=& 2+4s \\ y&=&2 -3s \\z&=& 1 + 2s\end{array} \right.
    Attached Thumbnails Attached Thumbnails line of intersection of planes-schnittgerade_3ebenen.gif  
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  3. #3
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    ic..so thats a line of intersection of 2 planes.so the unit vector which is parallel to that line is \frac{4}{\sqrt{29}},\frac{-3}{\sqrt{29}},\frac{2}{\sqrt{29}} ? so to deduce, any line which is parallel to another line will have the same direction vector and a point which lies on it.. so the equation of the line of intersection is [a,b,c]+s[4,-3,2]..since we are not given any points which is on that line..pls correct me if im wrong..
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  4. #4
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    Quote Originally Posted by ashes View Post
    ic..so thats a line of intersection of 2 planes.so the unit vector which is parallel to that line is \frac{4}{\sqrt{29}},\frac{-3}{\sqrt{29}},\frac{2}{\sqrt{29}} ?...... Right!

    so to deduce, any line which is parallel to another line will have the same direction vector and a point which lies on it........ Right!

    so the equation of the line of intersection is [a,b,c]+s[4,-3,2]..since we are not given any points which is on that line........ Right!

    pls correct me if im wrong.. Not necessary! You got it.
    ..
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