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# Are these vectors in the non-negative orthant of some $R^K$?

Given three sets of vectors $S_i=\left(V_{i1},V_{i2},\ldots,V_{in}\right),i=1,2,3$, s.t. the vectors within a set are pair-wise orthogonal and $V_{ij}\cdot V_{kl}\geq 0$ $\forall i,j,k,l$. Also, $\sum_jV_{ij}=w$ $\forall i$ where $w$ is some unit vector and $V_{ij}\cdot w=V_{ij}\cdot V_{ij}$ $\forall i,j$. It is given that the vectors in $S_i,S_j$ $\forall i\neq j$ lie in $R_{\geq 0}^k$ for some $k$. Can all the three sets of vectors be accommodated in $R_{\geq 0}^K$ for some $K$?
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