Determine a vector for a $3\times 3$ matrix so it becomes positive orthogonal

Question

Two orthonormal vectors in $\mathbb{R}^3$ are given:

$v_1=(-\frac{1}{2}\sqrt{2}, 0, \frac{1}{2}\sqrt{2})$ and $v_2=(0,1,0)$

Let $Q$ be a $3\times 3$ matrix with the first, second and third column be $v_1$, $v_2$ and $v_3$ respectively.

Determine $v_3$ so $Q$ becomes a positive orthogonal matrix.

I'm stuck. Any hints? I know that $det(Q)=1$ for it to be positive orthogonal, but I'm not sure how to get there from the given information.

Answer 1

Hint: The vector cross product of any two vectors is orthogonal to each.