Let $\lambda$ be an interger. If the shortest distance between the lines $x-\lambda=2 y-1=-2 z$ and $x=y+ 2 \lambda=z-\lambda$ is $\frac{\sqrt{7}}{2 \sqrt{2}}$, then the value...

The vector equation of the plane passing through the intersection of the planes $\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=1$ and $\vec{r} \cdot(\hat{i}-2 \hat{j})=-2$, and the point $(1,0,2)$ is :

Let $\mathrm{a}, \mathrm{b} \in \mathrm{R}$. If the mirror image of the point $\mathrm{P}(\mathrm{a}, 6$, 9) with respect to the line $\frac{x-3}{7}=\frac{y-2}{5}=\frac{z-1}{-9}$ is $(20, b,-a-9)$, then...

The equation of the planes parallel to the plane x – 2y + 2z – 3 = 0 which are at unit distance from the...

Let the plane $a x+b y+c z+d=0$ bisect the line joining the points $(4,-3,1)$ and $(2,3,-5)$ at the right angles. If $a, b, c, d$...

A vector $\vec{a}$ has components $3 p$ and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about...

Let $P$ be a plane $l x+m y+n z=0$ containing the line, $\frac{1-x}{1}=\frac{y+4}{2}=\frac{z+2}{3}$. IF plane $P$ divides the line segment $A B$ joining points $A(-3,-6,1)$...

Let the position vectors of two points $P$ and $Q$ be $3 \hat{i}+\hat{j}+2 \hat{k}$ and $\hat{i}+2 \hat{j}-4 \hat{k}$, respectively. Let $R$ and $S$ be two...

Let $(\lambda, 2,1)$ be a point on the plane which passes through the point $(4,-2,2)$. If the plane is perpendicular to the line joining the...

If for $\mathrm{a}>0$, the feet of perpendiculars from the points $\mathrm{A}(\mathrm{a},-2 \mathrm{a}, 3)$ and $\mathrm{B}(0,4,5)$ on the plane $l \mathrm{x}+\mathrm{my}+\mathrm{nz}=0$ are points $\mathrm{C}(0,-\mathrm{a},-1)$ and D...