Question

Are the four points $A(1,-1,1),B(-1,1,1),C(1,1,1)$ and $D(2,-3,4)$ are coplaner? Justify your answer.

Answer 1

The points are $A(1,-1,1),B(-1,1,1),C(1,1,1),D(2,-3,4)$.

The equation of plane formed by the points A, B, C is given by :

$\left|\begin{array}{ccc}x-1& y+1& z-1\\ -2& 2& 0\\ 0& 2& 0\end{array}\right|=0$

$(x-1)\left(0\right)-(y+1)\left(0\right)+(z-1)(-4)=0$

$-4z+4=0$

$z=1$

If point D lies on the plane then it must satisfy the equation of the plane.

But, $4\ne 1$

Hence, the given points are not coplanar.