Orthogonals,Orthonormals,bases.

Question

Given: $B = ((2/3, 1/3,2/3),(1/3,2/3,-2/3),(2/3,-2/3,-1/3))$ How can I check if B is an orthonormal base of $R^3$?

I thought about first checking if these vectors are a base. Then checking if they are an orthogonal base. And then confirming that they can also be an orthonormal base.

Answer 1

let's first check if they are orthonormal or not. $$(2/3, 1/3,2/3).(1/3,2/3,-2/3)=0$$ $$(1/3,2/3,-2/3).(2/3,−2/3,−1/3)=0$$ and similarly $$(2/3, 1/3,2/3).(2/3,−2/3,−1/3)=0$$.

Also $$||(2/3,−2/3,−1/3)||=||(1/3,2/3,-2/3)||=||(2/3, 1/3,2/3)||=1$$. So they are orthonormal. All it remains to be seen if they are linearly independent or not which I hope you can do.

Answer 2

If these $3$ vectors are orthogonal then they form a basis of $\Bbb R^3$ so it suffices to verify that they are orthogonal by computing their dot product and we verify that their norm is equal to $1$.