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! 

The vector product is a common operation in modern vector analysis.


*

*It gives you a vector with length
$$\lVert a \times b \rVert = \lVert a \rVert \, \lVert b \rVert \sin \angle(a, b)$$

*Then $c = a \times b$ has the properties
$0 = c \cdot a = c \cdot b$, using the scalar product, so it is orthogonal to both $a$ and $b$. However this and the above is true as well for $-c$.

*So the last property is that $a, b, c$ are arranged like in the right-hand-rule. 


 (Source)
All three properities can be derived from the connection to the vector product with the determinant:
$$
u \cdot (a \times b) = 
u_i \epsilon_{ijk} a_j b_k =
\epsilon_{ijk} u_i a_j b_k =
\text{det}(u, a, b)
$$
The significance of the vector product comes from its use in physics, especially mechanics and electro dynamics.
E.g. angular momentum $L = r \times p$ or Lorentz force $F_L = q v \times B$.
