I think you need help with thedefinitions. A single vector is NOT a "basis", othonormal or not, unless you have a vector space of dimension 1 which I do not believe is the case here. I think you are asking how to go from the basis {(1,-2,2), (2,1,0), (-2,0,1)} of to the orthonormal basis {(1/3,-2/3,2/3), 1/sqr(5)(2,1,0),1/3sqr(5)(-2,4,5)}.

I wrote out an entire long, complicated, explanation of the "Gram-Schmidt" orthogonalization process, to create a basis of vectors "orthogonal" (perpendicular) to one another from a general basis and then, applying it to this example, started getting "0"s! The first two vectors are already orthogonal to the each other, and the first to the second but not the second to the third. We can go ahead and "normalize" the first two. (1,-2,2) has length . To get a vector of length 1 in the same direction as (1,-2,2), divide by 3: (1/3, -2/3, 2/3). That is the first vector in your new "orthonormal" basis. To get a vector of length 1 in the same direction as (2,1,0), divide by its length: . is the "normalized" vector in the same direction as (2,1,0).

The third is the hardest because (-2,0,1) is not already perpendicular to (2,1,0) and so not to . To construct a vector that is, we need to divide (-2,0,1) into two "components", one parallel to , the other perpendicular. And, since is in the same direction as (2,1,0) we can get simplify the calculations by using (2,1,0). To get the one parallel to (2, 1,0), "project" (-2,0,1) onto (2,1,0). I'm sure you already have a formula for that. The projection of vector on vector is . Here, [tex](-2,0,1)\cdot(2,1,0)= -4[tex] and we have already seen that |(2,1,0)|= 5 so the projection of (-2,0,1) onto (2,1,0) is (-4/5)(2,1,0)= (-8/5, -4/5, 0). That is parallel to (2,1,0) and so (-2,0,1)- (-8/5, -4/5, 0)= (-2/5, 4/5, 1) is orthogonal to (2,1,0)- you can check the dot product: . We might also check that so (-2/5, 4/5, 1) is still perpendicular to (1,-2,2).

Now, we need only normalize (-2/5, 4/5, 1). It's length is . Dividing (-2/5, 4/5, 1) by gives , the third vector in your orthonormal basis.