question about orthogonal vector Let say I have 2 vectors $(1, 0, 0)$ and $(0, 2, 0)$, and I want to find a third vector that is orthogonal to both of them. I can do a cross product and get $(0, 0, 2)$. However, I know there are infinite vector in the following form $(0, 0, x)$ where $x$ $\in$ R that are orthogonal to the other two. My question is what is the difference between the orthogonal vector results from cross product and any other orthogonal vector? Why the cross product gives only 1 specific orthogonal vector? What is the significance of this vector? Thank you! 

If $\vec{a}$ and $\vec{b}$ are two vectors then the vector $\vec{a} \times \vec{b}$ is another vector perpendicular, or normal, the plane formed by $\vec{a}$ and $\vec{b}$. As you rightly suspected, there are an infinite number of vectors normal to the plane formed by $\vec{a}$ and $\vec{b}$. They are all of the form $\alpha(\vec{a} \times \vec{b})$, where $\alpha$ is a scalar. Since all of these "point" to the same direction, it is customary to choose a vector that has unit magnitude. It is given by 
\begin{equation}
\vec{c} = \frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|},
\end{equation}
where $|\cdot|$ denotes the magnitude of a vector.
