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

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(A) Let the points be $A(1,2,2), B(2,1,2), C(2,2,z)$ and $D(1,1,1)$.For these points to be coplanar,the scalar triple product of vectors $\vec{AB}, \v

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

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(A) Let the points be $A(1,2,2), B(2,1,2), C(2,2,z)$ and $D(1,1,1)$.For these points to be coplanar,the scalar triple product of vectors $\vec{AB}, \vec{AC}$ and $\vec{AD}$ must be $0$.$\vec{AB} = (2-1, 1-2, 2-2) = (1, -1, 0)$$\vec{AC} = (2-1, 2-2, z-2) = (1, 0, z-2)$$\vec{AD} = (1-1, 1-2, 1-2) = (0, -1, -1)$The condition for coplanarity is $\left|\begin{array}{ccc} 1 & -1 & 0 \ 1 & 0 & z-2 \ 0 & -1 & -1 \end{array}ight| = 0$.Expanding along the first row: $1(0 - (-(z-2))) - (-1)(-1 - 0) + 0 = 0$$1(z-2) - 1 = 0$$z - 2 - 1 = 0 \Rightarrow z = 3$.Thus,Statement $1$ is false.Statement $2$ is a standard geometric property: the volume of a tetrahedron formed by $4$ coplanar points is $0$. Hence,Statement $2$ 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 the $4$ 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 the $4$ points $P, Q, R$ and $S$ are coplanar,then the volume of the tetrahedron $PQRS$ is $0$.