Summary: 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.