12.1 Videos Guide

12.1a

• Distance between two points in space

  • $d\left(P_{1},\ P_{2}\right)=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$

• Equation of a sphere with center $(h,\ k,\ l)$ and radius $r$

  • $(x-h)^{2}+(y-k)^{2}+(z-l)^{2}=r^{2}$

12.1b

Exercises:

• Show that the equation represents a sphere, and find its center and radius.
  $$x^{2}+y^{2}+z^{2}-2x-4y+8z=15$$

• Find an equation of a sphere if one of its diameters has endpoints $(5,\ 4,\ 3)$ and $(1,\ 6,\ -9)$.

• Describe in words the region of $\mathbb{R}^{3}$ represented by the equation $x^{2}+y^{2}=4$.