1.
A variable plane is at a constant perpendicular distance $p$ from the origin and meets the coordinate axes at points $A, B,$ and $C$. Prove analytically that the locus of the centroid of the tetrahedron $OABC$ is given by the equation $x^{-2} + y^{-2} + z^{-2} = 16p^{-2}$.
2.
Find the magnitude and the equations of the line of shortest distance between the lines $L_1: \frac{x-8}{3} = \frac{y+9}{-16} = \frac{z-10}{7}$ and $L_2: \frac{x-15}{3} = \frac{y-29}{8} = \frac{z-5}{-5}$.
3.
A variable plane passes through a fixed point $(a, b, c)$ and meets the coordinate axes at the points $A, B,$ and $C$. Show that the locus of the point common to the three planes drawn through $A, B,$ and $C$ parallel to the coordinate planes is $\frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 1$.
4.
Find the exact equation of the line which represents the reflection of the line $\frac{x-1}{9} = \frac{y-2}{-1} = \frac{z+3}{-3}$ in the plane $3x - 3y + 10z = 26$.
5.
Find the distance of the point $P(3, 8, 2)$ from the line $\frac{x-1}{2} = \frac{y-3}{4} = \frac{z-2}{3}$ measured parallel to the plane $3x + 2y - 2z + 15 = 0$.
6.
Using the properties of scalar triple products, prove that the lines given by $\frac{x-a+d}{\alpha-\delta} = \frac{y-a}{\alpha} = \frac{z-a-d}{\alpha+\delta}$ and $\frac{x-b+c}{\beta-\gamma} = \frac{y-b}{\beta} = \frac{z-b-c}{\beta+\gamma}$ are strictly coplanar.
7.
Find the equations of the lines which bisect the angles between the two lines given by $\frac{x-1}{2} = \frac{y+1}{-2} = \frac{z}{1}$ and $\frac{x-1}{1} = \frac{y+1}{2} = \frac{z}{-2}$.
8.
A line $L$ passes through the origin and makes angles $\alpha, \beta, \gamma$ with the coordinate planes $x=0, y=0, z=0$ respectively. Prove that $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 2$.
9.
Find the geometric condition under which the three planes $x=cy+bz$, $y=az+cx$, and $z=bx+ay$ intersect exactly in a single line.
10.
A plane passes through a fixed point $(p, q, r)$ and cuts the coordinate axes in points $A, B,$ and $C$. Find the locus of the center of the sphere passing through the origin $O$ and the points $A, B, C$.
11.
Find the equation of the line passing through the origin which intersects the two lines $L_1: \frac{x-3}{1} = \frac{y-5}{2} = \frac{z-7}{3}$ and $L_2: \frac{x+1}{2} = \frac{y+2}{1} = \frac{z+3}{4}$.
12.
A tetrahedron has vertices at $O(0,0,0), A(1,2,1), B(2,1,3),$ and $C(-1,1,2)$. Find the exact acute angle between the two triangular faces $OAB$ and $ABC$.
13.
In a cube of edge length '$a$', find the shortest distance between the main diagonal of the cube and a face diagonal that does not intersect the main diagonal.
14.
Find the coordinates of the two possible points on the line $\frac{x+2}{3} = \frac{y+1}{2} = \frac{z-3}{2}$ that are situated at an exact distance of $3\sqrt{2}$ units from the point $(1, 2, 3)$.
15.
Find the equation of the plane which contains the line of intersection of the planes $x+2y+3z-4=0$ and $2x+y-z+5=0$, and which is perpendicular to the plane $5x+3y-6z+8=0$.
