Three Dimensional Geometry

Consider a sphere passing through the origin and the points $(2,1,-1),(1,5,-4),(-2,4,-6)$. Consider the following statements: The sphere passes through the point $(0,4,0)$. The point $(1,1,1)$ is at a distance of 5 unit…

Let $L_1$ be the line of intersection of the planes given by the equation- $2 x+3 y+z=4 \text { and } x+2 y+z=5 .$ Let $L_2$ be the line passing through the point $P(2,-1,3)$ and parallel to $L_1$. Let $M$ denote the…

A straight line drawn from the point $P(1,3,2)$, parallel to the line $\frac{x-2}{1}=\frac{y-4}{2}=\frac{z-6}{1}$, intersects the plane $L_1: x-y+3 z=6$ at the point $Q$. Another straight line which passes through $Q$…

Let $\gamma \in \mathbb{R}$ be such that the lines $\mathrm{L}_1: \frac{x+11}{1}=\frac{\mathrm{y}+21}{2}=\frac{\mathrm{z}+29}{3}$ and $L_2: \frac{x+16}{3}=\frac{y+11}{2}=\frac{z+4}{\gamma}$ intersect. Let $\mathrm{R}_1$…

Let $\mathbb{R}^3$ denote the three-dimensional space. Take two points $P=(1,2,3)$ and $Q=(4,2,7)$. Let $\operatorname{dist}(X, Y)$ denote the distance between two points $X$ and $Y$ in $\mathbb{R}^3$. Let…

Let $Q$ be the cube with the set of vertices ${(x_{1},x_{2},x_{3})\inℝ^{3}:x_{1},x_{2},x_{3}\in{0,1}}$. Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$. Let $S$ be the…

Let $\alpha,\beta,\gamma,\delta$ be real numbers such that $\alpha^{2}+\beta^{2}+\gamma^{2}\neq0$ and $\alpha+\gamma=1$. Suppose the point $(3,2,-1)$ is the mirror image of the point $(1,0,-1)$ with respect to the plane…

Let $P$ be the image of the point$(3,1,7)$ with respect to the plane $x-y+z=3.$ Then the equation of the plane passing through $P$ and containing the straight line $\frac{x}{1}=\frac{y}{2}=\frac{z}{1}$ is

Consider a pyramid$OPQRS$ located in the first octant $(x\geq0,y\geq0,z\geq0)$ with$O$, as origin, and$OP$ and,$OR$ along the $x‐axis$ and the $y‐axis$, respectively. The base$OPQR$ of the pyramid is a square…

In $R^{3}$, consider the planes $P_{1}:y=0$ and $P_{2}:x+z=1.$ Let $P_{3}$, be a plane, different from $P_{1}$ and $P_{2}$, which passes through the intersection of $P_{1}$ and $P_{2}$. If the distance of the point…

A line $l⁡$ passing through the origin is perpendicular to the lines $l⁡_{1}:(3+\text{t})\text{i}^+(-1+2\text{t})\text{j}^+(4+2\text{t})\text{k}^,-\infty<\text{t}<\infty$…

The point $P$ is the intersection of the straight line joining the points $Q(2,3,5)$ and $R(1,-1,4)$ with the plane $5 x-4 y-$ $z=1$. If $S$ is the foot of the perpendicular drawn from the point $T(2,1,4)$ to $Q R$,…