If the plane 2x + y – 5z = 0 is rotated about its line of intersection with the plane 3x – y + 4z...

Let the points on the plane P be equidistant from the points (.4, 2, 1) and (2, .2, 3). Then the acute angle between the...

If the shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{\lambda}$ and $\frac{x-2}{1}=\frac{y-4}{4}=\frac{z-5}{5}$ is $\frac{1}{\sqrt{3}}$, then the sum of all possible values of $\lambda$ is :

Let $P_1: \bar{r} .(2 \hat{i}+\hat{j}-3 \hat{k})=4$ be a plane. Let $P_2$ be another plane which passes through the points $(2,-3,2)(2$, $-2,-3)$ and $(1,-4,2)$. If the...

Let $d$ be the distance between the foot of perpendiculars of the points $P(1,2-1)$ and $Q(2,-1,3)$ on the plane $-x+y+z=1$. Then $d^2$ is equal to...

Let $P$ be the plane passing through the intersection of the planes $\vec{r} \cdot(\hat{i}+3 \hat{j}-\hat{k})=5$ and $\vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=3$, and the point $(2,1,-2)$. Let the...

Let the mirror image of the point $(a, b, c)$ with respect to the plane $3 x-4 y+12 z+19=0$ be $(\alpha-6, \beta, \gamma)$. If $...

If $\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}, \vec{b}=3 \hat{i}+3 \hat{j}+\hat{k}$ and $\vec{c}=c_1 \hat{i}+c_2 \hat{j}+c_3 \hat{k}$ are coplanar vectors and $\vec{a} \cdot \vec{c}=5, \vec{b} \perp \vec{c}$, then $122\left(c_1+c_2\right. +\mathrm{c}_3$...

Let the plane $P: \vec{r} \cdot \vec{a}=d$ contain the line of intersection of two planes $\vec{r} \cdot(\hat{i}+3 \hat{j}-\hat{k})=6$ and $\vec{r} \cdot(-6 \hat{i}+5 \hat{j}-\hat{k})=7$. If the...