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Determine all planes orthogonal to (1,1,1)

  1. Feb 8, 2005 #1
    Hello, could someone please give me some help with the following question?

    Q. Determine all planes (in R³) orthogonal to the vector (1,1,1).

    This is how I started off but I am not really sure how I need to go about solving this problem. I begin(by somewhat assuming that the vector (1,1,1) is perpendicular to the relevant planes) by writing the point normal form of planes with the n = (1,1,1) so I get (1,1,1).(x-p)=0.

    With x = (x,y,z) I get down to [tex]x + y + z = \left( {1,1,1} \right) \bullet \mathop p\limits_\~[/tex]. With other questions I am given the point P so the dot product of the vectors 'n' and 'p' can be found. With this one the situation is different because I need to find all planes which are orthogonal to the vector (1,1,1). I thought about letting the vector p = (f,g,h) but that doesn't seem right. Could someone help me out with this one? Any help is appreciated.

    Edit: My program for using Tex seems a little screwy at the moment so I had to fix part sof my post. X and p are supposed to denote vectors.
    Last edited: Feb 8, 2005
  2. jcsd
  3. Feb 8, 2005 #2


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    If (x,y,z) is a general point on the plane and (x0,y0,z0) is a fixed point on the plane, then (x-x0)i+ (y-0j)+ (z-z0)k is a vector in the plane. If <1,1,1> is perpendicular to the plane, it is perpendicular to all vectors in the plane so <1,1,1>.<x-x0,y-y0,z-z0>= 0.

    In fact, by the time you are expected to do a problem like this, you should already have learned that any plane can be written as Ax+ By+ Cz= D where <A, B, C> is a vector perpendicular to the plane
  4. Feb 8, 2005 #3
    WHen a plane is orthogonal to a vector, what is that vector called??

    THat gives you something about the family of planes and since we know that the scalar equation of a plane is Ax+By+Cz=D where (A,B,C) is the NORMAL vector to the plane and it doesn'd D is the intercept, but since in this case we want a family D can be anything!
  5. Feb 8, 2005 #4
    Thanks for clarifying the point about D. I usually find the value of D by calculating n.p but since a family of planes is needed I can just denote p by by (f,g,h) or something similar.
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