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

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!

• $v$ and $-v$ are distinct vectors, and either will work to give an orthonormal basis (so these would be two distinct bases).
– Dave
Commented Jan 21, 2020 at 18:47
• The two bases will have different orientations: one is a rotation of the standard basis while the other is a reflection.
– amd
Commented Jan 22, 2020 at 9:52

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.