  Let $n,k$ denote integers with $n>2k\geq 6$. Let $\mathbb{F}_q$ denote a
finite field with $q$ elements, and let $V$ denote a vector space over
$\mathbb{F}_q$ that has dimension $n$. The projective geometry $P_q(n)$ is the
partially ordered set consisting of the subspaces of $V$; the partial order is
given by inclusion. For the Grassmann graph $J_q(n,k)$ the vertex set consists
of the $k$-dimensional subspaces of $V$. Two vertices of $J_q(n,k)$ are
adjacent whenever their intersection has dimension $k-1$. The graph $J_q(n,k)$
is known to be distance-regular. Let $\partial$ denote the path-length distance
function of $J_q(n,k)$. Pick two vertices $x,y$ in $J_q(n,k)$ such that
$1<\partial(x,y)<k$. The set $P_q(n)$ contains the elements $x,y,x\cap y,x+y$.
In our main result, we describe $x\cap y$ and $x+y$ using only the graph
structure of $J_q(n,k)$. To achieve this result, we make heavy use of the
Euclidean representation of $J_q(n,k)$ that corresponds to the second largest
eigenvalue of the adjacency matrix.
