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Normal vector of a plane

  1. Feb 12, 2008 #1

    tony873004

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    From the book's example, the normal vectors of the planes x+y+z=1 and x-2y+3z=1 are <1,1,1> and <1, -2, 3>.

    Although the book doesn't mention how it got those normal vectors from the equations, it's rather obvious. But the first homework problem has the plane equation = 0 instead of equal 1. Can I still just pull the coefficients of x, y, z and form a normal vector? i.e. If the equation of the plane is x+z=0, then is the normal vector <1,0,1>?
     
  2. jcsd
  3. Feb 12, 2008 #2
    Yup, you can always pull the coefficients off for the normal because that term after the = sign doesn't change the slopes of the plane -- it will just determine intercepts and points through which the plane passes
     
  4. Feb 12, 2008 #3

    tony873004

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    thanks! The book failed to explain that.
     
  5. Feb 12, 2008 #4

    tony873004

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    I better post the full problem because I'm stuck again.

    Find the parametric and symmetric equations of the line of intersection of the planes x+y+z=1 and x+z=0.

    I got the normal vectors, <1,1,1> and <1,0,1> and their cross product <1,0,-1> or i-k.

    I set z to 0 and got x=0, y=1, z=0.

    How do I form parametric equation out of this?? I know it's x=t, y=1, z=-t because this problem is nearly identical to one from lecture. But how did he do that step?

    This would make the symmetric equations x/1=y-1/0=z/-1. But I can't divide by 0, can I?
     
  6. Feb 13, 2008 #5

    HallsofIvy

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    I'm not sure why you are worrying about vectors. I would just solve the two equations for two of the variables in terms of the third. Subtracting the third equation from the first, we get y= 1 From the third equation, z= -x. Taking x itself as parameter, we have x= t, y= 0, z= -t.

    As for the "symmetric" equations, yes, the fact that y is constant causes a problem! The only "symmetric" are z= -x, y= 1.
     
  7. Feb 18, 2008 #6

    tony873004

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    A belated thank you, Halls. I didn't notice your response until now. I guess I stopped monitoring this thread after I turned in the homework. I also got z=-x, y=1. Thanks for confirming that for me.
     
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