Three Dimensional Geometry

Let $P(3,2,3),Q(4,6,2)$ and $R(7,3,2)$ be the vertices of $\Delta\,\mathrm{PQR}$. Then, the angle $\angle\,\mathrm{QPR}$ is

Let the plane $P$contain the line $2x+y-z-3=0=5x-3y+4z+9$ and be parallel to the line $\frac{x+2}{2}=\frac{3-y}{-4}=\frac{z-7}{5}$. Then the distance of the point $A\left(8,-1,-19\right)$ from the plane $P$ measured…

Let $P$be the plane passing through the points $\left(5,3,0\right),\left(13,3,-2\right)$ and $\left(1,6,2\right)$. For$\alpha\,\in\,\mathbb{N}$, if the distance of the points$A\left(3,4,\alpha\,\right)$ and…

Let $PQR$ be a triangle with $R(-1,4,2)$. Suppose $M(2,1,2)$ is the mid point of $PQ$. The distance of the centroid of $\Delta\,PQR$ from the point of intersection of the line $\frac{x-2}{0}=\frac{y}{2}=\frac{z+3}{-1}$…

Let $O$ be the origin and the position vector of $A$ and $B$ be $2\hat{i}+2\hat{j}+\hat{k}$ and $2\hat{i}+4\hat{j}+4\hat{k}$ respectively. If the internal bisector of $\angle\,AOB$ meets the line $AB$ at $C$, then the…

The square of the distance of the image of the point $(6,1,5)$ in the line $\frac{x-1}{3}=\frac{y}{2}=\frac{z-2}{4}$, from the origin is _________

Let a line passing through the point $(-1,2,3)$intersect the lines $L_{1}:\frac{x-1}{3}=\frac{y-2}{2}=\frac{z+1}{-2}$ at $M(\alpha\,,\beta\,,\gamma\,)$ and $L_{2}:\frac{x+2}{-3}=\frac{y-2}{-2}=\frac{z-1}{4}$ at…

The distance of the point $Q(0,2,-2)$ form the line passing through the point $P(5,-4,3)$ and perpendicular to the…

Let $P$ be the point $(10,-2,-1)$ and $Q$ be the foot of the perpendicular drawn from the point $R(1,7,6)$ on the line passing through the points $(2,-5,11)$ and $(-6,7,-5)$. Then the length of the line segment $P Q$ is…

The lines $\frac{x-2}{2}=\frac{y}{-2}=\frac{z-7}{16}$ and $\frac{x+3}{4}=\frac{y+2}{3}=\frac{z+2}{1}$ intersect at the point $P$. If the distance of $P$ from the line $\frac{x+1}{2}=\frac{y-1}{3}=\frac{z-1}{1}$ is $l$,…

Let P be the image of the point $\mathrm{Q}(7,-2,5)$ in the line $\mathrm{L}: \frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4}$ and $\mathrm{R}(5, \mathrm{p}, \mathrm{q})$ be a point on $L$. Then the square of the area of…

Consider a line $\mathrm{L}$ passing through the points $\mathrm{P}(1,2,1)$ and $\mathrm{Q}(2,1,-1)$. If the mirror image of the point $\mathrm{A}(2,2,2)$ in the line $\mathrm{L}$ is $(\alpha, \beta, \gamma)$, then…