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

[Benny]

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 $$x + y + z = \left( {1,1,1} \right) \bullet \mathop p\limits_\~$$. 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.

 

[HallsofIvy]

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

 

[stunner5000pt]

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!

 

[Benny]

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.