Short question about a third vector and orthonormal basis in $\mathbb R^3$

Question

I have a task that asks to complete this set of vectors
$\begin{bmatrix}
2/3 \\
2/3 \\
-1/3
\end{bmatrix},$ $\begin{bmatrix}
2/3 \\
-1/3 \\
2/3
\end{bmatrix}$

to an orthonormal basis in $\mathbb R^3$

I solved a system that finds the third vector and obtained $2$ solutions (two vectors). But the thing is these two vectors are $\{v, -v\}$.

My question: do I have actually only one vector that allows to construct the basis(as the second vector is just the opposite) or is it two distinct vectors that allow to construct two different orthonormal bases? Thx in advance!

Answer 1

If $\{v_1,v_2,v_3\}$ is an orthonormal basis of $\mathbb R^3$, then so is $\{v_1,v_2,-v_3\}$. So, it is natural that you got two answers. It could not have been otherwise.