Let the plane $2 x+3 y+z+20=0$ be rotated through a right angle about its line of intersection with the plane $x-3 y+5 z=8$. If the...

If the two lines $\ell_1: \frac{x-2}{3}=\frac{y+1}{-2}, z=2$ and $\ell_2: \frac{x-1}{1}=\frac{2 y+3}{\alpha}=\frac{z+5}{2}$ perpendicular, then an angle between the lines $l_2$ and $\ell_3: \frac{1-\mathrm{x}}{3}=\frac{2 \mathrm{y}-1}{-4}=\frac{\mathrm{z}}{4}$ is:

Let $\theta$ be the angle between the vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ where $|\overrightarrow{\mathrm{a}}|=4,|\overrightarrow{\mathrm{~b}}|=3 \theta \in\left(\frac{\pi}{4}, \frac{\pi}{3}\right)$. Then

If the shortest distance between the line $\vec{r}=(-\hat{i}+3 \hat{k})+\lambda(\hat{i}-a \hat{j})$ and $\vec{r}=(-\hat{j}+2 \hat{k})+\mu(\hat{i}-\hat{j}+\hat{k})$ is $\sqrt{\frac{2}{3}}$ then the integral value of is equal to

Let $Q$ be the mirror image of the point $P(1,0,1)$ with respect to the plane $S: x+y+z=5$. If a line $L$ passing through $(1,-1,-1)$, parallel...

Let a line having direction ratios $1,-4,2$ intersect the lines $\frac{x-7}{3}=\frac{y-1}{-1}=\frac{z+2}{1}$ and $\frac{x}{2}=\frac{y-7}{3}=\frac{z}{1}$ at the point $A$ and $B$. Then $(A B)^2$ is equal to...