A(1, 2, 3), B(3, 8, 2) & C(-1, 0, - 5).
\(\vec {AB} = \vec{a_2} - \vec{a_1} = (3 - 1)\hat i + (8 -2 )\hat j + (2 - 3)\hat k = 2\hat i + 6\hat j - \hat k\)
\(\vec {AC} = \vec {a_3} - \vec {a_1} = (-1-1)\hat i + (0 - 2) \hat j + (-5- 3)\hat k = -2 \hat i - 2\hat j - 8 \hat k\)
\(\vec {AB} \times \vec {AC} = \begin{vmatrix} \hat i &\hat j&\hat k\\2&6&-1\\-2&-2&-8\end{vmatrix}\)
\(= \hat i (-48 - 2) - \hat j (-16 - 2) + \hat k (-4 + 12)\)
\(= -50\hat i + 18\hat j + 8\hat k\)
\(\therefore \) equation of plane
\((\vec r - \vec a).(\vec {AB} \times \vec{AC}) = 0\)
⇒ \((\vec r - (\hat i + 2\hat j + 3\hat k)).(-50\hat i + 18 \hat j - 18 \hat k) = 0\)
⇒ \([\vec r - (\hat i + 2\hat j + 3\hat k)].(-25\hat i + 9\hat j + 4\hat k) = 0\)
which is vector equation of required plane.
