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In this paper we study two directions of extending the classical Erd\Hos-Ko-Rado theorem which states that any family of $k$-element subsets of theset $[n] = \{1,\ldots,n\}$ in which any two sets intersect, has cardinality atmost ${n-1\choose k-1}$.  In the first part of the paper we study the families of $\{0,\pm1\}$-vectors. Denote by $\mathcal L_k$ the family of all vectors $\mathbf v$from $\{0,\pm 1\}^n$ such that $\langle\mathbf v,\mathbf v\rangle = k$. For any$k$, most $l$ and sufficiently large $n$ we determine the maximal size of thefamily $\mathcal V\subset \mathcal L_k$ such that for any $\mathbf v,\mathbfw\in \mathcal V$ we have $\langle \mathbf v,\mathbf w\rangle\ge l$. We findsome exact values of this function for all $n$ for small values of $k$.  In the second part of the paper we study cross-intersecting pairs offamilies. We say that two families are $\mathcal A, \mathcal B$ are\textit{$s$-cross-intersecting}, if for any $A\in\mathcal A,B\in \mathcal B$ wehave $|A\cap B|\ge s$. We also say that a set family $\mathcal A$ is {\it$t$-intersecting}, if for any $A_1,A_2\in \mathcal A$ we have $|A_1\cap A_2|\get$. For a pair of nonempty $s$-cross-intersecting $t$-intersecting families$\mathcal A,\mathcal B$ of $k$-sets, we determine the maximal value of$|\mathcal A|+|\mathcal B|$ for $n$ sufficiently large.

Intersection theorems for $\{0,\pm 1\}$-vectors and $s$-cross-intersecting families