Question

Derive the equation of a plane passing through three non-collinear points both in the vector and cartesian form .

Answer 1

$\text{Let}A\left(\overline{a}\right)=(x_1,y_1,z_1)$
$B=\left(\overline{b}\right)(x_2,y_2,z_2)$
$C=\left(\overline{c}\right)(x_3,y_3,z_3)$ be three non collinear points
Let $P\left(\overline{r}\right)=(x,y,z)$ be any point in the plane

$\text{Vector equation}:\Rightarrow P,A,B,C\text{are coplanar}\Rightarrow \overline{AP},\overline{AB},\overline{AC}\text{are coplanar}\Rightarrow [\overline{AP},\overline{AB},\overline{AC}]=0\Rightarrow [\overline{r}-\overline{a}\overline{b}-\overline{a}]=0\Rightarrow (\overline{r}-\overline{a}).(\overline{b}-\overline{a})\times (\overline{c}-\overline{a})=0$

$\text{Scalar equation}\Rightarrow \left|\begin{array}{ccc}x-x_1& y-y_1& z-z_1\\ x_2-x_1& y_2-y_1& z_2-z_1\\ x_3-x_1& y_3-y_1& z_3-z_1\end{array}\right|=0$