Three-Dimensional Coordinate Systems

(requires JavaScript)

1. Which of the points $P\left(6,2,3\right)$, $Q\left(-5,-1,4\right)$, and $R\left(0,3,8\right)$ is closest to the $xz\text{-plane}$? Which point lies in the $yz\text{-plane}$?
   $Q$; $R$.
2. Describe and sketch a surface in ${ℝ}^3$ represented by the equation $x+y=2$.
   A vertical plane that intersects the $xy\text{-plane}$ in the line $y=2-x$, $z=0$.
3. Find an equation of the sphere with center $\left(2,-6,4\right)$ and radius $5$. Describe its intersection with each of the coordinate planes.
   ${\left(x-2\right)}^2+{\left(y+6\right)}^2+{\left(z-4\right)}^2=25$
   circle of radius $3$ for $z=0$, circle of radius $\sqrt{21}$ for $x=0$, and empty set for $y=0$.
4. Show that the following equation is that of a sphere, and find its center and radius.

   $4x^2+4y^2+4z^2-8x+16y=1$

5. Find an equation of the sphere if one of its diameters has endpoints $\left(2,1,4\right)$ and $\left(4,3,10\right)$.
6. Describe in words the region of ${ℝ}^3$ represented by the equation $y=-4$.
   A plane parallel to the $xz\text{-plane}$ and $4$ units to the left of it.
7. Describe in words the region of ${ℝ}^3$ represented by the inequality $x^2+y^2+z^2>2z$.
8. Write inequalities describing the solid upper hemisphere of the sphere of radius $2$ centered at the origin.