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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?

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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 $Q$.

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  • $\begingroup$ No reason to specialize to $\mathbb{R}$. This is right over any field (maybe characteristic not $2$) using the standard inner product $\sum u_i v_i$. $\endgroup$ – David E Speyer May 29 '13 at 15:46

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