Question

Show that the four points (0, −1, −1), (4, 5, 1), (3, 9, 4) and (−4, 4, 4) are coplanar and find the equation of the common plane.

Answer 1

The equation of the plane passing through the points (0, −1, −1), (4, 5, 1) and (3, 9, 4) is given by

$$\begin{gathered} \left|\begin{array}{ccc}x-0& y+1& z+1\\ 4-0& 5+1& 1+1\\ 3-0& 9+1& 4+1\end{array}\right|=0 \\ \Rightarrow \left|\begin{array}{ccc}x& y+1& z+1\\ 4& 6& 2\\ 3& 10& 5\end{array}\right|=0 \\ \Rightarrow 10x-14\left(y+1\right)+22\left(z+1\right)=0 \\ \Rightarrow 5x-7\left(y+1\right)+11\left(z+1\right)=0 \\ \Rightarrow 5x-7y+11z+4=0 \\ \text{Substituting the last point (-4, 4, 4) (it means x = -4; y = 4; z = 4) in this plane equation, we get} \\ 5\left(-4\right)-7\left(4\right)+11\left(4\right)+4=0 \\ \Rightarrow -48+48=0 \\ \Rightarrow 0=0 \\ \text{So, the plane equation is satisfied by the point (-4, 4, 4).} \\ \text{So, the given points are coplanar and the equation of the common plane (as we already found) is} \\ 5x-7y+11z+4=0. \end{gathered}$$