Question

Find the equation of the plane passing through the following points.
(i) (2, 1, 0), (3, −2, −2) and (3, 1, 7)
(ii) (−5, 0, −6), (−3, 10, −9) and (−2, 6, −6)
(iii) (1, 1, 1), (1, −1, 2) and (−2, −2, 2)
(iv) (2, 3, 4), (−3, 5, 1) and (4, −1, 2)
(v) (0, −1, 0), (3, 3, 0) and (1, 1, 1)

Answer 1

(i) The equation of the plane passing through points (2, 1, 0), (3, −2, −2) and (3, 1, 7) is given by

$$\begin{gathered} \left|\begin{array}{ccc}x-2& y-1& z-0\\ 3-2& -2-1& -2-0\\ 3-2& 1-1& 7-0\end{array}\right|=0 \\ \Rightarrow \left|\begin{array}{ccc}x-2& y-1& z-0\\ 1& -3& -2\\ 1& 0& 7\end{array}\right|=0 \\ \Rightarrow -21\left(x-2\right)-9\left(y-1\right)+3z=0 \\ \Rightarrow -21x+42-9y+9+3z=0 \\ \Rightarrow -21x-9y+3z+51=0 \\ \Rightarrow 21x+9y-3z=51 \\ \Rightarrow 7x+3y-z=17 \end{gathered}$$

(ii) The equation of the plane passing through points (−5, 0, −6), (−3, 10, −9) and (−2, 6, −6) is given by

$$\begin{gathered} \left|\begin{array}{ccc}x+5& y-0& z+6\\ -3+5& 10-0& -9+6\\ -2+5& 6-0& -6+6\end{array}\right|=0 \\ \Rightarrow \left|\begin{array}{ccc}x+5& y& z+6\\ 2& 10& -3\\ 3& 6& 0\end{array}\right|=0 \\ \Rightarrow 18\left(x+5\right)-9y-18\left(z+6\right)=0 \\ \Rightarrow 2\left(x+5\right)-y-2\left(z+6\right)=0 \\ \Rightarrow 2x+10-y-2z-12=0 \\ \Rightarrow 2x-y-2z-2=0 \end{gathered}$$

(iii) The equation of the plane passing through points (1, 1, 1), (1, −1, 2) and (−2, −2, 2) is given by

$$\begin{gathered} \left|\begin{array}{ccc}x-1& y-1& z-1\\ 1-1& -1-1& 2-1\\ -2-1& -2-1& 2-1\end{array}\right|=0 \\ \Rightarrow \left|\begin{array}{ccc}x-1& y-1& z-1\\ 0& -2& 1\\ -3& -3& 1\end{array}\right|=0 \\ \Rightarrow 1\left(x-1\right)-3\left(y-1\right)-6\left(z-1\right)=0 \\ \Rightarrow x-1-3y+3-6z+6=0 \\ \Rightarrow x-3y-6z+8=0 \end{gathered}$$

(iv) The equation of the plane passing through points (2, 3, 4), (−3, 5, 1) and (4, −1, 2) is given by

$$\begin{gathered} \left|\begin{array}{ccc}x-2& y-3& z-4\\ -3-2& 5-3& 1-4\\ 4-2& -1-3& 2-4\end{array}\right|=0 \\ \Rightarrow \left|\begin{array}{ccc}x-2& y-3& z-4\\ -5& 2& -3\\ 2& -4& -2\end{array}\right|=0 \\ \Rightarrow -16\left(x-2\right)-16\left(y-3\right)+16\left(z-4\right)=0 \\ \Rightarrow \left(x-2\right)+\left(y-3\right)-\left(z-4\right)=0 \\ \Rightarrow x+y-z=1 \end{gathered}$$

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

$$\begin{gathered} \left|\begin{array}{ccc}x-0& y+1& z-0\\ 3-0& 3+1& 0-0\\ 1-0& 1+1& 1-0\end{array}\right|=0 \\ \Rightarrow \left|\begin{array}{ccc}x-0& y+1& z-0\\ 3& 4& 0\\ 1& 2& 1\end{array}\right|=0 \\ \Rightarrow 4x-3\left(y+1\right)+2z=0 \\ \Rightarrow 4x-3y+2z=3 \end{gathered}$$