4.
\((1)\)椭圆\( \dfrac{x^{2}}{9}+ \dfrac{y^{2}}{2}=1\)的焦点为\(F\)\({\,\!}_{1}\),\(F\)\({\,\!}_{2}\),点\(P\)在椭圆上,若\(|\)\(PF\)\({\,\!}_{1}|=4\),则\(∠\)\(F\)\({\,\!}_{1}\)\(PF\)\({\,\!}_{2}\)的大小为__________.
\((2)\)如果椭圆\( \dfrac{{x}^{2}}{36}+ \dfrac{{y}^{2}}{9}=1 \)的弦被点\((4,2)\)平分,则这条弦所在的直线方程
\((3)\)在正方体\(ABCD-{{A}_{1}}{{B}_{1}}{{C}_{1}}{{D}_{1}}\)中,\(M\)、\(N\)分别是\(CD\)、\(C{{C}_{1}}\)的中点,则异面直线\({{A}_{1}}M\)与\(DN\)所成角的大小是____________。
\((4)\)已知\(A\;\;,\;\;B\;,\;\;C \)三点不共线,\(O\)为平面\(ABC\)外一点,若由向量\(\overrightarrow{OP}=\dfrac{1}{5}\overrightarrow{OA}+\dfrac{2}{3}\overrightarrow{OB}+\lambda \overrightarrow{OC}\)确定的点\(P\)与\(A\;\;,\;\;B\;,\;\;C \)共面,那么\(\lambda =\) .