Embibe Experts Solutions for Chapter: Three Dimensional Geometry, Exercise 2: Level 2

Distance between the point $P\left(-2,\frac{-11}{2},-4\right)$ and the point of intersection of lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-{\displaystyle \frac{9}{2}}}{2}=\frac{z}{1}$ is

If $\lambda$ is the shortest distance between the lines $\frac{4x-3}{8}=\frac{2y+1}{6}=\frac{7z+6}{28}$ and $\frac{4x-3}{5}=\frac{2y+1}{6}=\frac{7z-6}{7}$, then the value of $91\lambda -30$ is

The angle between two diagonals of a cube will be

The plane through the intersection of planes $x+y+z=1$ and $2x+3y-z+4=0$ and parallel to $Y-$axis, also passes through the point

A line $l$ passing through origin is perpendicular to the lines

$l_1:\overrightarrow{r}=\left(3+t\right)\hat{\mathrm{i}}+\left(-1+2t\right)\hat{\mathrm{j}}+\left(4+2t\right)\hat{\mathrm{k}}$

$l_2:\overrightarrow{r}=\left(3+2s\right)\hat{\mathrm{i}}+\left(3+2s\right)\hat{\mathrm{j}}+\left(2+s\right)\hat{\mathrm{k}}$

If the co-ordinates of the point in the first octant on $l_2$ at a distance of $\sqrt{17}$ from the point of intersection of $l$ and $l_1$ are $\left(a, b, c\right)$, then $18\left(a+b+c\right)$ is equal to ___ .

Let $P$ be a plane passing through the points $(1,0,1),(1,-2,1)$ and $(0,1,-2).$ Let a vector $\overrightarrow{\mathrm{a}}=\alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}}$ be such that $\overrightarrow{a}$ is parallel to the plane $P$, perpendicular to $\left(\hat{i}+2\hat{j}+3\hat{k}\right)$ and $\overrightarrow{a}·(\hat{i}+\hat{j}+2\hat{k})=2$, then $(\alpha -\beta +\gamma {)}^2$ equals______.

A point on the plane determined by the points $A(1, 1, -1), B(2, -1, 0)$ and $C(-1, 0, 2)$ among the following is

Let $Q$ be the foot of the perpendicular from the point $P(7,-2,13)$ on the plane containing the lines $\frac{x+1}{6}=\frac{y-1}{7}=\frac{z-3}{8}$ and $\frac{x-1}{3}=\frac{y-2}{5}=\frac{z-3}{7}$
Then $(PQ{)}^2,$ is equal to ______.