Three Dimensional Geometry

Let the coordinates of the points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ be $(1,8,4)$, $(0,-11,4)$ and $(2,-3,1)$ respectively. What are the coordinates of the point $D$ which is the foot of the perpendicular from A on BC…

A line through $(1,-1,2)$ with direction ratios $ \lt 3,2,2 \gt $ meets the plane $x+2 y+3 z=18$. What is the point of intersection of line and plane?

For the following two (02) items: A plane $P$ is parallel to the line having direction ratios $\langle 1, 3, 2 \rangle$ and contains the line of intersection of the planes $6x + 4y - 5z = 2$ and $x - 2y + 3z = 0$. Which…

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 $D$?

What is the angle between the lines $2 x=3 y=-z$ and $6 x=-y=-4 z ?$

Paragraph: Let $\mathrm{L}: x+y+z+4=0=2 x-y-z+8$ be a line and $\mathrm{P}: x+2 y+3 z+1=0$ be a plane Question: What is the point of intersection of $L$ and $P$ ?

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 is the equation of the line $L$ ?

If $p$ is the perpendicular distance from origin to the plane passing through $(1,0,0),(0,1,0)$ and $(0,0,1)$, then what is $3 p^2$ equal to?

Passage: $P(-2,-3,5)$, $Q(4,-1,5)$, $R(6,-4,8)$ and $S(2,-6,10)$ are four points. Question: What is $(PQ^2+2QS^2-2PR^2)$ equal to ?

Paragraph: Consider two lines whose direction ratios are $(2,-1,2)$ and $(k, 3,5)$. They are inclined at an angle $\frac{\pi}{4}$. Question: What is the value of $k$ ?

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 $P$ ?

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?