If the foot of the perpendicular from point $(4,3,8)$ on the line $\mathrm{L}_1: \frac{\mathrm{x}-\mathrm{a}}{\ell}=\frac{\mathrm{y}-2}{3}=\frac{\mathrm{z}-\mathrm{b}}{4}, \ell \neq 0$ is $(3,5$, 7), then the shortest distance between...

Consider the line $L$ given by the equation $\frac{x-3}{2}=\frac{y-1}{1}=\frac{z-2}{1}$. Let $Q$ be the mirror image of the point $(2,3$, -1) with respect to $L$. Let...

If $(x, y, z)$ be an arbitrary point lying on a plane $P$ which passes through the point $(42,0,0),(0,42,0)$ and ( $0,0,42$ ), then the...

The lines $x=a y-1=z-2$ and $x=3 y-2=b z-2,(a b \neq 0)$ are coplanar, if .

The equation of the plane which contains the y-axis and passes through the point (1, 2, 3) is :

Let $\vec{a}=2 \hat{i}-3 \hat{j}+4 \hat{k}$ and $\vec{b}=7 \hat{i}+\hat{j}-6 \hat{k}$ If $\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})=-3$ then $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+\hat{\mathrm{k}})$ is equal...

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...