• Support PF! Buy your school textbooks, materials and every day products Here!

Find the basis of a plane

  • #1

Homework Statement


Let W be the plane
3x + 2y − z = 0 in ℝ3.
Find a basis for W perpendicular


Homework Equations





The Attempt at a Solution


I thought a basis for this plane could be generated just by letting x=0 and y=1, finding z and then doing the same thing but this time letting x=1 and y=0 and finding z. If you do that you get:

[0].........[1]
[1].........[0]
[2].........[3]

Apparently this is wrong, can anybody tell me what's wrong about this?
 

Answers and Replies

  • #2
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,523
737
You have just given two points on the plane or, interpreting them as vectors, the position vectors to those points. You want the normal vector [3,2,-1]. At least that's what I think the question is asking for.
 
  • #3
360
0

Homework Statement


Let W be the plane
3x + 2y − z = 0 in ℝ3.
Find a basis for W perpendicular
I think the question is asking for a basis of the orthogonal complement of W, usually denoted [itex] W^{\perp} [/itex] and read "W perp." So you wouldn't be looking for a basis for W, but for the set of all vectors perpendicular to W.

EDIT: LCKurtz beat me to it.
 
  • #4
I think the question is asking for a basis of the orthogonal complement of W, usually denoted [itex] W^{\perp} [/itex] and read "W perp." So you wouldn't be looking for a basis for W, but for the set of all vectors perpendicular to W.

EDIT: LCKurtz beat me to it.
so are you saying that after getting the two vectors that i got solving for z, i should find any 2 other vectors that make the dot product with the first ones zero?
 
  • #5
33,314
5,006
No, I don't think that's what they're saying at all. W is the plane, a two-dimensional subspace of R3. W[itex]^\bot[/itex] ("W perp") is therefore a one-dimensional subspace of R3. The normal to the plane is in the same direction as W[itex]^\bot[/itex].
 
  • #6
No, I don't think that's what they're saying at all. W is the plane, a two-dimensional subspace of R3. W[itex]^\bot[/itex] ("W perp") is therefore a one-dimensional subspace of R3. The normal to the plane is in the same direction as W[itex]^\bot[/itex].
ohh ok so then the answer is going to be just the vector of the plane itself?

[3,2,-1]
 
  • #7
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,523
737
ohh ok so then the answer is going to be just the vector of the plane itself?

[3,2,-1]
That isn't the "vector of the plane". It is a vector perpendicular to the plane.
 

Related Threads on Find the basis of a plane

Replies
1
Views
879
  • Last Post
Replies
7
Views
932
  • Last Post
Replies
4
Views
22K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
12
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
10
Views
1K
  • Last Post
Replies
9
Views
1K
  • Last Post
Replies
10
Views
2K
Top