I'm trying to prove that (u x v) is orthogonal to both u and v. Is it a sufficient proof to simply demonstrate that the dot product of u and (u x v) is equal to zero because due to the properties of the cross product, the previous expression is equivalent to the dot product of (u x u) and v. Since the cross product of u with itself is obviously 0, we can see that u is orthogonal to (u x v). I would repeat this same process for v.
Is that a sufficient proof?
The logic is valid. Since $x \cdot (y \times z) = z \cdot (x \times y)$, and $x \times x = 0$ for all $x, y, z \in \mathbb{R}^3$, we do indeed have $u \cdot (u \times v) = v \cdot (u \times v) = 0$.
The question of whether the proof is sufficent or not depends a little on context. I'm guessing that you're taking some kind of course? If so, make sure you're allowed to use these facts. Specifically, check to see if they're in the lecture notes. Many lecturers don't like it when you pull facts off the internet in an introductory course.