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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

Vedclass · Accepted Answer

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

Vedclass · Answer

(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 $0$.

Similar Questions

If $x \cdot a = 0, x \cdot b = 0$ and $x \cdot c = 0$ for some non-zero vector $x$,then the true statement is

The vectors $i + 2j + 3k$,$\lambda i + 4j + 7k$,and $-3i - 2j - 5k$ are collinear if $\lambda$ equals:

The value of $\alpha$,so that the volume of the parallelepiped formed by $\hat{i}+\alpha \hat{j}+\hat{k}$,$\hat{j}+\alpha \hat{k}$,and $\alpha \hat{i}+\hat{k}$ becomes maximum,is

The volume of the parallelepiped having vertices at $O \equiv (0,0,0)$, $A \equiv (2,-2,1)$, $B \equiv (5,-4,4)$, and $C \equiv (1,-2,4)$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module

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 $0$.