Section 1
Analytic Geometry in Three Dimensions
Find the drstance between the pairs of points in Exercises 1-4, $(0,0,0)$ and $(2,-1,-2)$ 2. $(-1,-1,-1)$ and $(1,1,1)$
$(1,1,0)$ and $(0,2,-2)$
$(3,8,-1)$ and $(-2,3,-6)$
What is the shortest distance from the point $\left(x_{4}, y_{0}, 2\right)$ to(a) the $x$ y-plane? (b) the $x$-axis?
SHore that the thangle with vertices $(1,2,3),(4,0,3)$, and $(3,6,4)$ has a right angle.
Find the angle $A$ in the trungle with vertices $A=(2,-1,-1), A=(0,1,-2)$, and $C=(1,-3,1)$.
Show that the triangle with vertices $(1,2,3),(1,3,4)$, and $(0,3,3)$ is cquilateral.
Find the area of the triangle with vertices $(1,1,0),(1,0,1)$, and $(0,1,1)$.
What is the detance from the ongin to the point $\left(1_{2} 1_{\ldots \ldots .}\right.$ I) in $\mathbb{R}^{4}$ ?
What is the distance from the point $(1,1, \ldots .+1)$ in $n-4 p a c e$ to the closest point an the a taxis?
In Eretcises 12-23, describe (and sketch if possible) the set of points in $\mathrm{R}^{3}$ that satisfy the given equation or incquatify. $2=2$
$y \geq-1$
$z=x$
$x+y=1$
$x^{2}+y^{2}+z^{2}=4$
$(x-1)^{2}+(y+2)^{2}+(z-3)^{2}=4$.$
$x^{2}+y^{2}+z^{2}=22$
$y^{2}+z^{2} \leq 4$
$x^{2}+z^{2}=4$
$2=y^{2}$
$z \geq \sqrt{x^{2}+y^{2}}$
$x+2 y+3 z=6$
In Exercises 24-32, describe (and sketch if possible) the set points in $\mathrm{R}^{3}$ that satisfy the given pair of equations of inequal $\left\{\begin{array}{l}x=1 \\ y=2\end{array}\right.$
$\left\{\begin{array}{l}x=1 \\ y=z\end{array}\right.$
$\left\{\begin{array}{l}x^{2}+y^{2}+z^{2}=4 \\ z=1\end{array}\right.$
$\left\{\begin{array}{l}x^{2}+y^{2}+z^{2}=4 \\ x^{2}+y^{2}+z^{2}=4 x\end{array}\right.$
$\left\{\begin{array}{l}x^{2}+y^{2}+z^{2}=4 \\ x^{2}+z^{2}=1\end{array}\right.$
$\left\{\begin{array}{l}x^{2}+y^{2}=1 \\ z=x\end{array}\right.$
$\left\{\begin{array}{l}y \geq x \\ z \leq y\end{array}\right.$
$\left\{\begin{array}{l}x^{2}+y^{2} \leq 1 \\ z \geq y\end{array}\right
$\left\{\begin{array}{l}x^{2}+y^{2}+z^{2} \leq 1 \\ \sqrt{x^{2}+y^{2}} \leq 2\end{array}\right.$
In Eversises 33-36, specify the boundary and the interior of the plane sets 5 whose points $(x, y)$ satisfy the given conditions. Is 5 open, closed, or neither? $0<x^{2}+y^{2}<1$
$x \geq 0, \quad y<0$
$|x|+|y| \leq 1$
In Exercises 37-40, specify the boundary and the interior of the sets $S$ in 3-space whose points $\left(x_{,}, y_{,}\right.$z) satisfy the given conditions. Is $S$ epen, closed, or neither?$1 \leq x^{2}+y^{2}+2^{2} \leq 4$
$x \geq 0, \quad y>1, \quad z<2$
$(x-2)^{2}+(y-z)^{2}=0$
$x^{2}+y^{2}<1 . y+2>2$