The shortest distance between the lines $\frac{x+2}{1}=\frac{y}{-2}=\frac{z-5}{2}$ and $\frac{x-4}{1}=\frac{y-1}{-2}=\frac{z+3}{0}$ is

Let $\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}$ and $\vec{c}=2 \hat{i}-\hat{j}+4 \hat{k}$. If a vector $\vec{d}$ satisfies $\vec{d} \times \vec{b}=\vec{c} \times \vec{b}$ and $...

Let $A B C D$ be a quadrilateral. If $E$ and $F$ are the mid points of the diagonals $A C$ and $B D$ respectively...

If the equation of the plane passing through the line of intersection of the planes 2x - y + z = 3, 4x - 3y...

Let S be the set of all values of $\lambda$, for which the shortest distance between the lines $\frac{x-\lambda}{0}=\frac{y-3}{4}=\frac{x+6}{1}$ and $\frac{x+\lambda}{3}=\frac{y}{-4}=\frac{x-6}{0}$ is 13 . Then...

The distance of the point $(-1,2,3)$ from the plane $\vec{r} \cdot(\hat{i}-2 \hat{j}+3 \hat{k})=10$ parallel to the line of the shortest distance between the lines $...

Let $\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=2 \hat{i}-2 \hat{j}-2 \hat{k}$ and $\vec{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$. If $\vec{d}$ is a vector perpendicular to both $\vec{b}$ and $\vec{c}$ and...

Let a line $I$ pass through the origin and be perpendicular to the lines $\mathrm{I}_1: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-1 \hat{\mathrm{j}}-7 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}), \lambda \in \mathbb{R}$ and $...

Let the plane $x+3 y-2 z+6=0$ meet the co-ordinate axes at the points A,B,C. It the orthocentre of the triangle ABC is $\left(\alpha, \beta, \frac{6}{7}\right)$...

Let $\lambda_1, \lambda_2$ be the values of $\lambda$ for which the points $\left(\frac{5}{2}, 1, \lambda\right)$ and $(-2,0,1)$ are at equal distance from the plane $...