How to find a vector that is perpendicular to every vector in a linear subspace?

Click For Summary
SUMMARY

The discussion centers on finding a unit vector that is perpendicular to every vector in the linear subspace of R³ defined by the equation x - y + z = 0. The subspace represents a plane in three-dimensional space. To determine a normal vector to this plane, one can directly extract the coefficients from the equation, leading to the normal vector <1, -1, 1>. Normalizing this vector yields the unit vector <1/√3, -1/√3, 1/√3>, which is perpendicular to all vectors in the specified subspace.

PREREQUISITES
  • Understanding of linear algebra concepts, particularly subspaces and normal vectors.
  • Familiarity with vector normalization techniques.
  • Knowledge of geometric interpretations of linear equations in R³.
  • Basic proficiency in using vector operations such as the cross product.
NEXT STEPS
  • Study the properties of normal vectors in linear algebra.
  • Learn about vector normalization and its applications.
  • Explore the geometric interpretation of planes and subspaces in R³.
  • Investigate the use of the cross product in finding perpendicular vectors.
USEFUL FOR

Students studying linear algebra, particularly those preparing for exams involving vector spaces and geometric interpretations in three-dimensional space.

ohjeezus1
Messages
1
Reaction score
0

Homework Statement


Hi, i don't know if you can help me but i am currently studying for my finals and i have come across a question which i am very confused about. i have looked it up in books but there seems to be no answer there. the question is Write down a vector of length 1 that is perpendicular to every vector in the linear subspace of r3 described by x-y+z=0. If you could help me i would be very greatful! thank you.
I know that to find a normal vector which is perpendicular to another vector you use the cross product but i do not see how this will benefit me in this question as i am not given any vectors!
 
Physics news on Phys.org
A first step would be to find out which subspace we're talking about, i.e. write down what form the vectors of this subspace take.
 
The subspace is {<x, y, z> in R3 | x - y + z = 0}. What sort of a geometric object is this subspace?
 

Similar threads

Prove that ##S## is a subspace of ##V##
  • · Replies 2 ·
Replies
2
Views
2K
To find the component of a vector perpendicular to another
  • · Replies 20 ·
Replies
20
Views
4K
Given value of vectors a,b, b.c and a+(b×c), Find (c.a)
  • · Replies 5 ·
Replies
5
Views
2K
Finding Linearly Independent Vectors in Subspaces
  • · Replies 6 ·
Replies
6
Views
2K
Vectors: Proving the value of a•(b x c)
  • · Replies 21 ·
Replies
21
Views
8K
Proving Subspaces of Vector Spaces: Evaluating A Vector x
  • · Replies 3 ·
Replies
3
Views
2K
Proving Vector <a,1,1> is a Subspace in R3
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Closure in the subspace of linear combinations of vectors
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Smallest subspace if a plane and a line are passing through the origin
  • · Replies 2 ·
Replies
2
Views
2K
Locus of all points -- Imprecise question?
  • · Replies 12 ·
Replies
12
Views
4K