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?
Best Answer
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 $(0,0,1)$