Let us complete {(2,0,3)} to form an orthogonal basis. Consider the vector (3,0,−2). Since 2⋅3+0⋅0+3⋅(−2)=0, we conclude that the inner product of (2,0,3) and (3,0,−2) is zero, and hence the vectors (2,0,3) and (3,0,−2) are orthogonal. Further, consider the vector (0,1,0). It follows that 2⋅0+0⋅1+3⋅0=0 and 3⋅0+0⋅1−2⋅0=0, and hence the vector (0,1,0) is orthogonal to the vectors (2,0,3) and (3,0,−2). Therefore, {(2,0,3),(3,0,−2),(0,1,0)} is an orthogonal basis of R3.
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