SOLUTION: Find an orthonormal basis of the plane x−4y−z=0.

Question 1160838: Find an orthonormal basis of the plane x−4y−z=0.

Answer by ikleyn(52814) (Show Source): You can put this solution on YOUR website!
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Find an orthonormal basis of the plane x−4y−z=0.
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Consider two vectors  V1 = (1,0,1)  and V2 = (2,1,-2).


You can manually check that both vectors V1 and V2 satisfy the given equation - 
- so, they belong to the plane described by this equation.


Next, it is clear that vectors V1 and V2 are linearly independent - hence, they form 
a basis in the plane described by the given equation.


The fact is that vectors V1 and V2 are orthogonal.
You may check it on your own.


To get vectors V1 and V2 orthonormal, we should divide each vector by its length.


Doing so, we get orthonormal vectors  ( ,,)  and  (,,).

Thus the problem is solved completely.

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