How can three vectors be orthogonal to each other? In a scenario, say that:

Vectors $\mathbf{U}$, $\mathbf{V}$ and $\mathbf{W}$ are all orthogonal such that the dot product between each of these $(\mathbf{UV}\;\mathbf{VW}\;\mathbf{WU})$ is equal to zero.

I imagine that for any potential vector space $\mathbf{R}$ this would only be possible in two situations.
1) $\mathbf{U}$, $\mathbf{W}$ and/or $\mathbf{V}$ is the zero vector.
2) $\mathbf{U}=(1, 0, 0)$, $\mathbf{V} = (0, 1, 0)$ and $\mathbf{W} = (0, 0, 1)$.
Is there any other situation where three vectors are all orthogonal to each other?
 A: There are infinitely many triple of non zero orthogonal vectors obtained by the three you have indicated by scaling of each one and rotations of the triple all togheter.
To construct any othogonal triple we can proceed as follows:


*

*choose a first vector $v_1=(a,b,c)$

*find a second vector orthogonal to $v_1$ that is e.g. $v_2=(-b,a,0)$

*determine the third by cross product $v_3=v_1\times v_2$
A: How about $v_1= (\frac{1}{\sqrt2}, \frac{1}{\sqrt2},0),v_2=(-\frac{1}{\sqrt2},\frac{1}{\sqrt2},0),v_3=(0,0,1)$ ?
Geometrically take any frame and rotate in any direction. 
A: Here is an example of just what you seek:

A: Think it like this: for a given vector $u\neq 0$ in $\mathbb{R}^3$, what is the space of all vectors perpendicular to it? It is a plane $P$ containing the origin, whose perpendicular direction is obviously $u$.
Now take a $v\neq 0$ in that plane. The space of all vectors perpendicular to $v$ is a new plane $P'$. In order to get a vector $w$ perpendicular to both $u,v$, we need $w\in P\cap P'$. What does $P\cap P'$ look like?
By the way, notice that $v$ is ANY non-zero vector, not only $(1,0,0), (0,1,0)$ or $(0,0,1)$
A: There are infinitely many possibilities. $(1,0,0), (0,1,-1)$ and $(0,1,1)$ is one example. 
A: The word "orthogonal" really just corresponds to the intuitive notion of vectors being perpendicular to each other. Draw out the unit vectors in the $x$, $y$ and $z$ directions respectively--those are one set of three mutually orthogonal (i.e. perpendicular) vectors, just like you observed. But if you rotate those three vectors together in any way that you like in 3D space, without changing the angles between them, then of course they will still remain orthogonal. 
Furthermore, if we scale these vectors by changing their lengths independently by any arbitrary nonzero factor, we will still end up with a set of orthogonal vectors, since the only thing that matters is the directions and not the lengths of the vectors. In this manner we end up with a description for an infinite family of orthogonal vectors, which hopefully makes it easy for you to convince yourself intuitively.
In a more general vector space, of course, this sort of pictorial intuition might no longer hold, but the idea of orthogonality can be easily generalised. That's the reason we define orthogonality abstractly and independent of the usual geometric notion of perpendicularity.
