SOLUTION: Determining an orthogonal basis for W = {(x, y, z); x - 2y + z = 0}.

Algebra ->  College  -> Linear Algebra -> SOLUTION: Determining an orthogonal basis for W = {(x, y, z); x - 2y + z = 0}.      Log On


   

Question 892544: Determining an orthogonal basis for W = {(x, y, z); x - 2y + z = 0}.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Choose any vector in W.
Set x=0, then -2y%2Bz=0
2y=z
If y=1, then z=2
(0,1,2)
.
.
.
Now use the dot product to find a perpendicular vector to this vector.
0%2Aa%2B1%2Ab%2B2%2Ac=0
Let a=1
then b=-2c
Let c=1, then b=-2
(1,-2,1)
.
.
.
Now take the cross product of those two vectors to find a mutually perpendicular vector to these two.
(0,1,2)X(1,-2,1)=(5,2,-1)
So then,
(0,1,2), (1,-2,1), and (5,2,-1) form an orthogonal basis of W.