Let $l\_1$ and $l\_2$ be two lines in $G(1,n)$, the Grassmannian of lines in n dimensional projective space.  

Suppose that their Plucker embeddings has dot product zero. Namely if $(x\_1, x\_2, \cdots, x\_N)$
and $(y\_1, y\_2, \cdots, y\_N)$ are their Plucker coordinates, $\sum\_{i=1}^N x\_i y\_i = 0$.
Is there anything special about $l\_1$ and $l\_2$, e.g. are they necessarily orthogonal to each other? 



Two lines with orthogonal Plucker embedding
In a real vector space you can say the following. The lines you refer to correspond to planes $P$, $Q$ in $\mathbb{R}^{n+1}$. The dot product of the Plucker coordinates is zero if and only if $P$ intersects the orthogonal complement of $Q$ nontrivially. This doesn't mean the lines are orthogonal. In fact they need not intersect, e.g. if $P$ is contained in the orthogonal complement to $Q$.


