Trong không gian Oxyz, cho 3 điểm \(A\,(1;0;3);\,\...

B

- Hướng dẫn giải

Gọi \(H\,(a;b;c)\) là trực tâm tam giác ABC thì \(\overrightarrow{AB};\,\,\overrightarrow{AC};\,\,\overrightarrow{AH}\) đồng phẳng

Ta có: \(\overrightarrow{AB}(-2;2;-2)=-2(1;-1;1);\,\,\overrightarrow{AC}=(-1;1;1);\)

\(\overrightarrow{CH}=(a;b-1;c-4);\,\,\overrightarrow{BH}=(a+1;b-2;c-1);\,\,\overrightarrow{AH}=(a-1;b;c-3)\)

Suy ra \(\left[ \overrightarrow{AB},\overrightarrow{AC} \right]=(4;4;0)=4(1;1;0)\)

Mặt khác

\(\left\{ \begin{array} {} \left[ \overrightarrow{AB},\overrightarrow{AC} \right]\overrightarrow{AH}=0 \\ {} \overrightarrow{CH}.\overrightarrow{AB}=0 \\ {} \overrightarrow{BH}.\overrightarrow{AC}=0 \\ \end{array} \right.\\\Leftrightarrow \left\{ \begin{array} {} a+b-1=0 \\ {} a-b+1+c-4=0 \\ {} -a-1+b-2+c-1=0 \\ \end{array} \right. \\\Leftrightarrow \left\{ \begin{array} {} a=\frac{1}{4} \\ {} b=\frac{3}{4} \\ {} c=\frac{7}{2} \\ \end{array} \right.\Rightarrow P=-\frac{1}{2}.\)