Section 1
Three-Dimensional Coordinate Systems
In Exercises $1-16,$ give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.$$x=2, \quad y=3$$
In Exercises $1-16,$ give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.$$x=-1, \quad z=0$$
In Exercises $1-16,$ give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.$$y=0, \quad z=0$$
In Exercises $1-16,$ give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.$$x=1, \quad y=0$$
In Exercises $1-16,$ give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.$$x^{2}+y^{2}=4, \quad z=0$$
In Exercises $1-16,$ give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.$$x^{2}+y^{2}=4, \quad z=-2$$
In Exercises $1-16,$ give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.$$x^{2}+z^{2}=4, \quad y=0$$
In Exercises $1-16,$ give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.$$y^{2}+z^{2}=1, \quad x=0$$
In Exercises $1-16,$ give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.$$x^{2}+y^{2}+z^{2}=1, \quad x=0$$
In Exercises $1-16,$ give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.$$x^{2}+y^{2}+z^{2}=25, \quad y=-4$$
In Exercises $1-16,$ give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.$$x^{2}+y^{2}+(z+3)^{2}=25, \quad z=0$$
In Exercises $1-16,$ give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.$$x^{2}+(y-1)^{2}+z^{2}=4, \quad y=0$$
In Exercises $1-16,$ give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.$$x^{2}+y^{2}=4, \quad z=y$$
In Exercises $1-16,$ give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.$$x^{2}+y^{2}+z^{2}=4, \quad y=x$$
In Exercises $1-16,$ give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.$$y=x^{2}, \quad z=0$$
In Exercises $1-16,$ give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.$$z=y^{2}, \quad x=1$$
In Exercises $17-24$ , describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.a. $x \geq 0, \quad y \geq 0, \quad z=0 \quad$ b. $x \geq 0, \quad y \leq 0, \quad z=0$
In Exercises $17-24$ , describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.a. $0 \leq x \leq 1 \quad$ b. $0 \leq x \leq 1, \quad 0 \leq y \leq 1$ c. $0 \leq x \leq 1, \quad 0 \leq y \leq 1, \quad 0 \leq z \leq 1$
In Exercises $17-24$ , describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.a. $x^{2}+y^{2}+z^{2} \leq 1 \quad$ b. $x^{2}+y^{2}+z^{2}>1$
In Exercises $17-24$ , describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.a. $x^{2}+y^{2} \leq 1, \quad z=0 \quad$ b. $x^{2}+y^{2} \leq 1, \quad z=3$ c. $x^{2}+y^{2} \leq 1,$ no restriction on $z$
In Exercises $17-24$ , describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.a. $1 \leq x^{2}+y^{2}+z^{2} \leq 4$ b. $x^{2}+y^{2}+z^{2} \leq 1, \quad z \geq 0$
In Exercises $17-24$ , describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.a. $x=y, \quad z=0 \quad$ b. $x=y, \quad$ no restriction on $z$
In Exercises $17-24$ , describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.a. $y \geq x^{2}, \quad z \geq 0 \quad$ b. $x \leq y^{2}, \quad 0 \leq z \leq 2$
In Exercises $17-24$ , describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.a. $z=1-y, \quad$ no restriction on $x$ b. $z=y^{3}, \quad x=2$
In Exercises $25-34$ , describe the given set with a single equation or with a pair of equations.The plane perpendicular to thea. $x$ -axis at $(3,0,0) \quad$ b. $y$ -axis at $(0,-1,0)$ c. $z$-axis at $(0,0,-2)$
In Exercises $25-34$ , describe the given set with a single equation or with a pair of equations.The plane through the point $(3,-1,2)$ perpendicular to thea. $x$ -axis b. $y$ -axis c. $z$ -axis
In Exercises $25-34$ , describe the given set with a single equation or with a pair of equations.The plane through the point $(3,-1,1)$ parallel to thea. $x y-p$ lane b. $y z-$ plane c. $x z-$ plane
In Exercises $25-34$ , describe the given set with a single equation or with a pair of equations.The circle of radius 2 centered at $(0,0,0)$ and lying in thea. $x y$ -plane b. $y z$ -plane c. $x z-$ plane
In Exercises $25-34$ , describe the given set with a single equation or with a pair of equations.The circle of radius 2 centered at $(0,2,0)$ and lying in thea. $x y$ -plane b. $y z$ -plane c. plane $y=2$
In Exercises $25-34$ , describe the given set with a single equation or with a pair of equations.The circle of radius 1 centered at $(-3,4,1)$ and lying in a plane parallel to thea. $x y$ -plane b. $y z$ -plane c. $x z$ -plane
In Exercises $25-34$ , describe the given set with a single equation or with a pair of equations.The line through the point $(1,3,-1)$ parallel to thea. $x$ -axis b. $y$ -axis c. $z$ -axis
In Exercises $25-34$ , describe the given set with a single equation or with a pair of equations.The set of points in space equidistant from the origin and the point $(0,2,0)$
In Exercises $25-34$ , describe the given set with a single equation or with a pair of equations.The circle in which the plane through the point $(1,1,3)$ perpendicular to the $z$ -axis meets the sphere of radius 5 centered at the origin
In Exercises $25-34$ , describe the given set with a single equation or with a pair of equations.The set of points in space that lie 2 units from the point $(0,0,1)$ and, at the same time, 2 units from the point $(0,0,-1)$
Write inequalities to describe the sets in Exercises $35-40$.The slab bounded by the planes $z=0$ and $z=1$ (planes included)
Write inequalities to describe the sets in Exercises $35-40$.The solid cube in the first octant bounded by the coordinate planes and the planes $x=2, y=2,$ and $z=2$
Write inequalities to describe the sets in Exercises $35-40$.The half-space consisting of the points on and below the $x y$ -plane
Write inequalities to describe the sets in Exercises $35-40$.The upper hemisphere of the sphere of radius 1 centered at the origin
Write inequalities to describe the sets in Exercises $35-40$.The (a) interior and (b) exterior of the sphere of radius 1 centered at the point $(1,1,1)$
Write inequalities to describe the sets in Exercises $35-40$.The closed region bounded by the spheres of radius 1 and radius 2 centered at the origin. (Closed means the spheres are to be included. Had we wanted the spheres left out, we would have asked for the open region bounded by the spheres. This is analogous to the way we use closed and open to describe intervals: closed means endpoints included, open means endpoints left out. Closed sets include boundaries; open sets leave them out.)
In Exercises $41-46,$ find the distance between points $P_{1}$ and $P_{2}.$$$P_{1}(1,1,1), \quad P_{2}(3,3,0)$$
In Exercises $41-46,$ find the distance between points $P_{1}$ and $P_{2}.$$$P_{1}(-1,1,5), \quad P_{2}(2,5,0)$$
In Exercises $41-46,$ find the distance between points $P_{1}$ and $P_{2}.$$$P_{1}(1,4,5), \quad P_{2}(4,-2,7)$$
In Exercises $41-46,$ find the distance between points $P_{1}$ and $P_{2}.$$$P_{1}(3,4,5), \quad P_{2}(2,3,4)$$
In Exercises $41-46,$ find the distance between points $P_{1}$ and $P_{2}.$$$P_{1}(0,0,0), \quad P_{2}(2,-2,-2)$$
In Exercises $41-46,$ find the distance between points $P_{1}$ and $P_{2}.$$$P_{1}(5,3,-2), \quad P_{2}(0,0,0)$$
Find the centers and radii of the spheres in Exercises $47-50$.$$(x+2)^{2}+y^{2}+(z-2)^{2}=8$$
Find the centers and radii of the spheres in Exercises $47-50$.$$(x-1)^{2}+\left(y+\frac{1}{2}\right)^{2}+(z+3)^{2}=25$$
Find the centers and radii of the spheres in Exercises $47-50$.$$(x-\sqrt{2})^{2}+(y-\sqrt{2})^{2}+(z+\sqrt{2})^{2}=2$$
Find the centers and radii of the spheres in Exercises $47-50$.$$x^{2}+\left(y+\frac{1}{3}\right)^{2}+\left(z-\frac{1}{3}\right)^{2}=\frac{16}{9}$$
Find equations for the spheres whose centers and radii are given in Exercises $51-54$ .$$\frac{\text { Center }}{(1,2,3)} \quad \frac{\text { Radius }}{\sqrt{14}}$$
Find equations for the spheres whose centers and radii are given in Exercises $51-54$ .$$\frac{\text { Center }}{(0,-1,5)} \quad \frac{\text { Radius }}{{2}}$$
Find equations for the spheres whose centers and radii are given in Exercises $51-54$ .$$\frac{\text { Center }}{\left(-1, \frac{1}{2},-\frac{2}{3}\right)} \quad \frac{\text { Radius }}{\frac{4}{9}}$$
Find equations for the spheres whose centers and radii are given in Exercises $51-54$ .$$\frac{\text { Center }}{(0,-7,0)} \quad \frac{\text { Radius }}{{7}}$$
Find the centers and radii of the spheres in Exercises $55-58$.$$x^{2}+y^{2}+z^{2}+4 x-4 z=0$$
Find the centers and radii of the spheres in Exercises $55-58$.$$x^{2}+y^{2}+z^{2}-6 y+8 z=0$$
Find the centers and radii of the spheres in Exercises $55-58$.$$2 x^{2}+2 y^{2}+2 z^{2}+x+y+z=9$$
Find the centers and radii of the spheres in Exercises $55-58$.$$3 x^{2}+3 y^{2}+3 z^{2}+2 y-2 z=9$$
Find a formula for the distance from the point $P(x, y, z)$ to thea. $x$ -axis b. $y$ -axis c. $z$ -axis
Find a formula for the distance from the point $P(x, y, z)$ to thea. $x y$ -plane b. $y z$ -plane c. $x z-$ plane
Find the perimeter of the triangle with vertices $A(-1,2,1),$ $B(1,-1,3),$ and $C(3,4,5)$
Show that the point $P(3,1,2)$ is equidistant from the points $A(2,-1,3)$ and $B(4,3,1)$ .
Find an equation for the set of all points equidistant from the planes $y=3$ and $y=-1 .$
Find an equation for the set of all points equidistant from the point $(0,0,2)$ and the $x y$ -plane.
Find the point on the sphere $x^{2}+(y-3)^{2}+(z+5)^{2}=4$ nearesta. the $x y$ -plane. b. the point $(0,7,-5)$
Find the point equidistant from the points $(0,0,0),(0,4,0),(3,0,0),$ and $(2,2,-3) .$