If the image of the point $\mathrm{P}(1,0,3)$ in the line joining the points $\mathrm{A}(4,7,1)$ and $\mathrm{B}(3,5,3)$ is $Q(\alpha, \beta, \gamma)$, then $\alpha+\beta+\gamma$ is equal to...

Let $\vec{a}=3 \hat{i}+\hat{j}-\hat{k}$ and $\vec{c}=2 \hat{i}-3 \hat{j}+3 \hat{k}$. If $\vec{b}$ is a vector such that $\vec{a}=\vec{b} \times \vec{c}$ and $|\vec{b}|^2=50$, then $\left|72-|\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}|^2\right|$ is equal to

Let the image of the point $\left(\frac{5}{3}, \frac{5}{3}, \frac{8}{3}\right)$ in the plane $x-2 y+z-2=0$ be $P$. If the distance of the point $Q(6-2, \alpha), \alpha>Q$...

Let the equation of plane passing through the line of intersection of the planes $x+2 y+a z=2$ and $x-y +z=3$ be $...

Let the plane $P$ contain the line $2 x+y-z-3=0=5 x-3 y+4 z+9$ and be parallel to the line $\frac{x+2}{2}=\frac{3-y}{-4}=\frac{z-7}{5}$. Then the distance of the point...

Let S be the set of all $(\lambda, \mu)$ for which the vectors $\lambda \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\mu \hat{\mathrm{k}}$ and $3 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$, where $\lambda-\mu=5$,...

Let the foot of perpendicular of the point $\mathrm{P}(3,-2-9)$ on the plane passing through the points $(-1,-2,-3),(9,3,4),(9,-2,1)$ be $\mathrm{Q}(\alpha, \beta, \gamma)$. Then the distance of...

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...