Embibe Experts Solutions for Chapter: Three Dimensional Geometry, Exercise 1: EXERCISE-1

Which one of the following statement is INCORRECT?

Consider the following $5$ statements

(I) There exists a plane containing the point $\left(1, 2, 3\right)$ and $\left(2, 3, 4\right)$ and perpendicular to the vector ${\overrightarrow{V}}_1=\widehat{\mathrm{i}}+\widehat{\mathrm{j}}-\widehat{\mathrm{k}}$

(II) There exist no plane containing the point $\left(1, 0, 0\right); \left(0, 1, 0\right); \left(0, 0, 1\right)$ and $\left(1, 1, 1\right)$

(III) If a plane  with normal vector $\overrightarrow{N}$ is perpendicular to $\overrightarrow{V}$, then $\overrightarrow{N}·\overrightarrow{V}=0$

(IV) If two planes are perpendicular then every line in one plane is perpendicular to every line on the other plane

(V) Let $P_1$ and $P_2$ are two perpendicular planes. If a third plane $P_3$ is perpendicular to $P_1$ then it must be either parallel or perpendicular or at an angle of $45°$ to $P_2$

Choose the correct alternative.

The distance between the parallel planes given by the equations $\overrightarrow{r}·\left(2\widehat{i}-2\widehat{j}+\widehat{k}\right)+3=0$ and $\overrightarrow{r}·\left(4\widehat{i}-4\widehat{j}+2\widehat{k}\right)+5=0$ is-

Which one of the following lines is parallel to the line

$L:\left(x,y,z\right)=\left(1,0,-2\right)+t\left(-1,3,0\right),t\in R$

The lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}$ and $\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}$ are coplanar, if

Given $A\left(1,-1,0\right); B\left(3,1,2\right); C\left(2,-2,4\right)$ and $D\left(-1,1,-1\right)$ which of the following points neither lie on $AB$ nor on $CD$?

For the line $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}$, which one of the following is incorrect?

Given the vertices $A\left(2,3,1\right), B\left(4,1,-2\right), C\left(6,3,7\right)$ and $D\left(-5,-4,8\right)$ of a tetrahedron. The length of the altitude drawn from the vertex $D$ is