Complete { (2, 0, 3)} to form an orthonormal basis of R^3
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Expert's answer
2021-12-07T13:34:32-0500
Orthogonal vector to one given vector (a,b,c) we take as (0,-c,b).
So for x1=(2,0,3) we have x2=(0,−3,0) with property x1⋅x2=2⋅0+0⋅(−3)+3⋅0=0 .
We serach third vector in the form x3=(a,b,c) . We must provide that x1⋅x3=2⋅a+0⋅b+3⋅c=2⋅a+3⋅c=0 and x2⋅x3=0⋅a+(−3)⋅b+0⋅c=(−3)⋅b=0
Thus we have the linear system:
{2a+3c=0−3b=0
This system has 3 unknowns: a,b,c, we may take one of them with arbitrary non zero value, let we assign c=2, then from the system we get a=-3,b=0 therefore x3=(-3,0,2).
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