Viết phương trình mặt phẳng $(P)$ đi qua $M(3;-1;-5)$ và vuông góc với $2$ mặt phẳng $(P_1): 3x - 2y + 2z =  - 7;    (P_2):   5x - 4y + 3z =  - 1$.
Ta có: $\left[ {\overrightarrow {{n_{{(P_1)}}}} ,\overrightarrow {{n_{{(P_2)}}}} } \right] = (-2;4;-2)// \overrightarrow{n_P}(1;-2;1) $.
Mặt phẳng (P) vuông góc với 2 mặt phẳng nên $(P)$ nhận $\overrightarrow{n_P} $ làm vectơ pháp tuyến

Do đó (P) : $2(x - 3) + 1.(y + 1) - 2(z + 5) = 0 \Rightarrow (P):\,\,2x + y - 2z-15=0$

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