Statement-1: If the points (1, 2, 2), (2, 1, | vedclass

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(D) Let the points be $A(1, 2, 2), B(2, 1, 2), C(2, 2, z)$ and $D(1, 1, 1)$.Four points are coplanar if the volume of the tetrahedron formed by them i

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Statement-$1$ is false,Statement-$2$ is true.

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(D) Let the points be $A(1, 2, 2), B(2, 1, 2), C(2, 2, z)$ and $D(1, 1, 1)$.Four points are coplanar if the volume of the tetrahedron formed by them is $0$. The volume $V$ is given by $\frac{1}{6} |[\vec{AB}, \vec{AC}, \vec{AD}]| = 0$.Vectors are:$\vec{AB} = (2-1)\hat{i} + (1-2)\hat{j} + (2-2)\hat{k} = \hat{i} - \hat{j} + 0\hat{k}$$\vec{AC} = (2-1)\hat{i} + (2-2)\hat{j} + (z-2)\hat{k} = \hat{i} + 0\hat{j} + (z-2)\hat{k}$$\vec{AD} = (1-1)\hat{i} + (1-2)\hat{j} + (1-2)\hat{k} = 0\hat{i} - \hat{j} - \hat{k}$The scalar triple product is the determinant:$\begin{vmatrix} 1 & -1 & 0 \ 1 & 0 & z-2 \ 0 & -1 & -1 \end{vmatrix} = 0$Expanding along the first row:$1(0 - (-(z-2))) - (-1)(-1 - 0) + 0 = 0$$1(z-2) - 1 = 0$$z - 2 - 1 = 0 \implies z = 3$.Since $z=3$ and not $2$,Statement-$1$ is false. Statement-$2$ is a standard geometric property of coplanar points,so it is true.

Statement-$1$: If the points $(1, 2, 2), (2, 1, 2), (2, 2, z)$ and $(1, 1, 1)$ are coplanar,then $z = 2$.
Statement-$2$: If $4$ points $P, Q, R$ and $S$ are coplanar,then the volume of the tetrahedron $PQRS$ is $0$.

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Statement-$1$: If the points $(1, 2, 2), (2, 1, 2), (2, 2, z)$ and $(1, 1, 1)$ are coplanar,then $z = 2$.Statement-$2$: If $4$ points $P, Q, R$ and $S$ are coplanar,then the volume of the tetrahedron $PQRS$ is $0$.