Three Dimensional Geometry

$A B$ and $C D$ are two line segments, where $A(2,3,0), B(6,9,0), C(-6,-9,0) . P$ and $Q$ are mid-point of $A B$ and $C D$, respectively and $L$ is the mid-point of $P Q$. Find the distance of $L$ from the plane $3 x+4…

If the lines$\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha\,}$and$\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta\,}$intersect, then the magnitude of the minimum value of $8\alpha\,\beta\,$is _____.

Suppose the line $\frac{x-2}{\alpha\,}=\frac{y-2}{-5}=\frac{z+2}{2}$ lies on the plane $x+3y-2z+\beta\,=0.$ Then $(\alpha\,+\beta\,)$ is equal to .

If the shortest distance between the lines$\vec{r_{1}}=\alpha\,\hat{i}+2\hat{j}+2\hat{k}+\lambda\,\left(\hat{i}-2\hat{j}+2\hat{k}\right),\lambda\,\in\,R,\alpha\,>0$and…

Let the equation of the plane passing through the line $x-2y-z-5=0=x+y+3z-5$and parallel to the line $x+y+2z-7=0=2x+3y+z-2$be $ax+by+cz=65$. Then the distance of the point $\left(a,b,c\right)$ from the plane…

Passage: A line $L$ passing through the point $(-1,2,-3)$ is perpendicular to the plane $P$ given by $2x+3y+z+5=0$. Question: What are the direction ratios of a line $M$ parallel to the plane $P$ ?

For the following two (02) items: Let $A(1, -1, 0)$, $B(-2, 1, 8)$ and $C(-1, 2, 7)$ are three consecutive vertices of a parallelogram $ABCD$. What is the fourth vertex $D$?

If a line in 3 dimensions makes angles $\alpha$, $\beta$ and $\gamma$ with the positive directions of the coordinate axes, then what is $\cos(\alpha+\beta)\cos(\alpha-\beta)$ equal to?

The sum of the direction cosines of $z$-axis is

A line passes through the points $(6,-7,-1)$ and $(2,-3,1)$. What are the direction ratios of the line?

The projections of a directed line segment on the coordinate axes are 12, 4, 3 respectively. What is the length of the line segment?

What should be the value of $k$ for which the equation $3 x^2+3 y^2+(k+1) z^2+x-y+z=0$ represents the sphere?