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

A line makes angles $\alpha ,\beta ,\gamma ,\delta$ with the four diagonals of a cube, then ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma +{\cos }^2\delta$ is equal to

Find the shortest distance between any two opposite edges of a tetrahedron formed by the planes $y+z=0,x+z=0,x+y=0,x+y+z=\sqrt{3}a$

If $\overrightarrow{r}$ represents the position vector of point $R$ in which the line $AB$cuts the plane $CDE,$ where the position vectors of points $A,B,C,D,E$ are respectively $\overrightarrow{a}=\hat{i}+2\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+\hat{j}+2\hat{k},\overrightarrow{c}=-4\overrightarrow{j}+4\hat{k},\overrightarrow{d}=2\hat{i}-2\hat{j}+2\hat{k}$ and $\overrightarrow{e}=4\hat{i}+\hat{j}+2\hat{k},$ then ${\overrightarrow{r}}^2$ is

Line $L_1$ is parallel to vector $\overrightarrow{\alpha }=-3\hat{i}+2\hat{j}+4\hat{k}$ and passes through a point $A\left(7,6,2\right)$ and line $L_2$ is parallel to a vector $\overrightarrow{\beta }=2\hat{i}+\hat{j}+3\hat{k}$ and passes through a point $B\left(5,3,4\right).$ Now, a line $L_3$ parallel to a vector $\overrightarrow{r}=2\hat{i}-2\hat{ȷ}-\hat{k}$ intersects the lines $L_1$ and $L_2$ at points $C$ and $D$ respectively, then $\left|4\overrightarrow{CD}\right|$ is equal to ?

The value of ${\sec }^3\theta ,$ where $\theta$ is the acute angle between the plane faces of a regular tetrahedron, is

$R$ and $r$ are the circum-radius and in-radius of a regular tetrahedron respectively in terms of the length $k$ of each edge. If $R^2+r^2=\frac{p}{q}{k}^2,$ where $p, q\in I$, then absolute minimum value of $p+q$ is

A line $L$ on the plane $2x+y-3z+5=0$ is at a distance $3$ unit from the point $P\left(1,2,3\right).$ A spider.starts moving from point $A$ and after moving $4$units  along the line $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-3}{-3}$ it reaches to point $P$ and from $P$, it jumps to line $L$ along the shortest distance and then moves $12$ units along the line $L$ to reach at point $B.$ The distance between points $A$ and $B$ is $k$ units, then the value of $k$ is

The line $\frac{x+6}{5}=\frac{y+10}{3}=\frac{z+14}{8}$ is the hypotenuse of an isosceles right angle triangle whose opposite vertex is $(7,2,4)$ Then the equation of remaining sides is/are -