A _______ coordinate system can be formed by passing a $z$-axis perpendicular to both the $x$-axis and the $y$-axis at the origin.

Ernest C.

Numerade Educator

The three coordinate planes of a three-dimensional coordinate system are the _______, the _______, and the _______ .

Taylor S.

Numerade Educator

The coordinate planes of a three-dimensional coordinate system separate the coordinate system into eight _______.

Ernest C.

Numerade Educator

The distance between the points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ can be found using the _______ _______ in Space.

Taylor S.

Numerade Educator

The midpoint of the line segment joining the points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ given by the Midpoint Formula in Space is _______ .

Ernest C.

Numerade Educator

A _______ is the set of all points $(x, y, z)$ such that the distance between $(x, y, z)$ and a fixed point $(h, k, j)$ is $r$.

Taylor S.

Numerade Educator

A _______ in _______ is the collection of points satisfying an equation involving $x$, $y$, and $z$.

Ernest C.

Numerade Educator

The intersection of a surface with one of the three coordinate planes is called a _______ of the surface.

Taylor S.

Numerade Educator

In Exercises 9 and 10, approximate the coordinates of the points.

Ernest C.

Numerade Educator

In Exercises 9 and 10, approximate the coordinates of the points.

Taylor S.

Numerade Educator

In Exercises 11-16, plot each point in the same three-dimensional coordinate system.

(a) $(2, 1, 3)$

(b) $(1, -1, -2)$

Ernest C.

Numerade Educator

In Exercises 11-16, plot each point in the same three-dimensional coordinate system.

(a) $(3, 0, 0)$

(b) $(-3, -2, -1)$

Taylor S.

Numerade Educator

In Exercises 11-16, plot each point in the same three-dimensional coordinate system.

(a) $(3, -1, 0)$

(b) $(-4, 2, 2)$

Ernest C.

Numerade Educator

In Exercises 11-16, plot each point in the same three-dimensional coordinate system.

(a) $(0, 4, -3)$

(b) $(4, 0, 4)$

Taylor S.

Numerade Educator

In Exercises 11-16, plot each point in the same three-dimensional coordinate system.

(a) $(3, -2, 5)$

(b) $(\frac{3}{2}, 4, -2)$

Ernest C.

Numerade Educator

In Exercises 11-16, plot each point in the same three-dimensional coordinate system.

(a) $(5, -2, 2)$

(b) $(5, -2, -2)$

Taylor S.

Numerade Educator

In Exercises 17-20, find the coordinates of the point.

The point is located three units behind the $yz$-plane, four units to the right of the $xz$-plane, and five units above the $xy$-plane.

Ernest C.

Numerade Educator

In Exercises 17-20, find the coordinates of the point.

The point is located seven units in front of the $yz$-plane,two units to the left of the $xz$-plane, and one unit below the $xy$-plane.

Taylor S.

Numerade Educator

In Exercises 17-20, find the coordinates of the point.

The point is located on the $x$-axis, eight units in front of the $yz$-plane.

Ernest C.

Numerade Educator

In Exercises 17-20, find the coordinates of the point.

The point is located in the $yz$-plane, one unit to the right of the $xz$-plane, and six units above the $xy$-plane.

Taylor S.

Numerade Educator

In Exercises 21-26, determine the octant(s) in which $(x, y, z)$ is located so that the condition(s) is (are) satisfied.

$x>0, y<0, z>0$

Ernest C.

Numerade Educator

In Exercises 21-26, determine the octant(s) in which $(x, y, z)$ is located so that the condition(s) is (are) satisfied.

$x<0, y>0, z<0$

Taylor S.

Numerade Educator

In Exercises 21-26, determine the octant(s) in which $(x, y, z)$ is located so that the condition(s) is (are) satisfied.

$z>0$

Ernest C.

Numerade Educator

$y<0$

Taylor S.

Numerade Educator

$xy<0$

Ernest C.

Numerade Educator

$yz<0$

Taylor S.

Numerade Educator

In Exercises 27-36, find the distance between the points.

$(0, 0, 0), \quad (5, 2, 6)$

Ernest C.

Numerade Educator

In Exercises 27-36, find the distance between the points.

$(1, 0, 0), \quad (7, 0, 4)$

Taylor S.

Numerade Educator

In Exercises 27-36, find the distance between the points.

$(3, 2, 5), \quad (7, 4, 8)$

Ernest C.

Numerade Educator

In Exercises 27-36, find the distance between the points.

$(4, 1, 5), \quad (8, 2, 6)$

Taylor S.

Numerade Educator

In Exercises 27-36, find the distance between the points.

$(-1, 4, -2), \quad (6, 0, -9)$

Ernest C.

Numerade Educator

In Exercises 27-36, find the distance between the points.

$(1, 1, -7), \quad (-2, -3, -7)$

Taylor S.

Numerade Educator

In Exercises 27-36, find the distance between the points.

$(0, -3, 0), \quad (1, 0, -10)$

Ernest C.

Numerade Educator

In Exercises 27-36, find the distance between the points.

$(2, -4, 0), \quad (0, 6, -3)$

Taylor S.

Numerade Educator

In Exercises 27-36, find the distance between the points.

$(6, -9, 1), \quad (-2, -1, 5)$

Ernest C.

Numerade Educator

In Exercises 27-36, find the distance between the points.

$(4, 0, -6), \quad (8, 8, 20)$

Taylor S.

Numerade Educator

In Exercises 37-40, find the lengths of the sides of the right triangle with the indicated vertices. Show that these lengths satisfy the Pythagorean Theorem.

$(0, 0, 2), \quad (-2, 5, 2), \quad (0, 4, 0)$

Ernest C.

Numerade Educator

In Exercises 37-40, find the lengths of the sides of the right triangle with the indicated vertices. Show that these lengths satisfy the Pythagorean Theorem.

$(2, -1, 2), \quad (-4, 4, 1), \quad (-2, 5, 0)$

Taylor S.

Numerade Educator

In Exercises 37-40, find the lengths of the sides of the right triangle with the indicated vertices. Show that these lengths satisfy the Pythagorean Theorem.

$(0, 0, 0), \quad (2, 2, 1), \quad (2, -4, 4)$

Ernest C.

Numerade Educator

$(1, 0, 1), \quad (1, 3, 1), \quad (1, 0, 3)$

Taylor S.

Numerade Educator

In Exercises 41-44, find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither.

$(1, -3, -2), \quad (5, -1, 2), \quad (-1, 1, 2)$

Ernest C.

Numerade Educator

In Exercises 41-44, find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither.

$(5, 3, 4), \quad (7, 1, 3), \quad (3, 5, 3)$

Taylor S.

Numerade Educator

In Exercises 41-44, find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither.

$(4, -1, -2), \quad (8, 1, 2), \quad (2, 3, 2)$

Ernest C.

Numerade Educator

$(1, -2, -1), \quad (3, 0, 0), \quad (3, -6, 3)$

Taylor S.

Numerade Educator

In Exercises 45-52, find the midpoint of the line segment joining the points.

$(0, 0, 0), \quad (3, -2, 4)$

Ernest C.

Numerade Educator

In Exercises 45-52, find the midpoint of the line segment joining the points.

$(1, 5, -1), \quad (2, 2, 2)$

Taylor S.

Numerade Educator

In Exercises 45-52, find the midpoint of the line segment joining the points.

$(3, -6, 10), \quad (-3, 4, 4)$

Ernest C.

Numerade Educator

In Exercises 45-52, find the midpoint of the line segment joining the points.

$(-1, 5, -3), \quad (3, 7, -1)$

Taylor S.

Numerade Educator

In Exercises 45-52, find the midpoint of the line segment joining the points.

$(-5, -2, 5), \quad (6, 3, -7)$

Ernest C.

Numerade Educator

In Exercises 45-52, find the midpoint of the line segment joining the points.

$(0, -2, 5), \quad (4, 2, 7)$

Taylor S.

Numerade Educator

In Exercises 45-52, find the midpoint of the line segment joining the points.

$(-2, 8, 10), \quad (7, -4, 2)$

Ernest C.

Numerade Educator

In Exercises 45-52, find the midpoint of the line segment joining the points.

$(9, -5, 1), \quad (9, -2, -4)$

Taylor S.

Numerade Educator

In Exercises 53-60, find the standard form of the equation of the sphere with the given characteristics.

Center: $(3, 2, 4);$ radius: $4$

Ernest C.

Numerade Educator

In Exercises 53-60, find the standard form of the equation of the sphere with the given characteristics.

Center: $(-3, 4, 3);$ radius: $2$

Taylor S.

Numerade Educator

In Exercises 53-60, find the standard form of the equation of the sphere with the given characteristics.

Center: $(5, 0, -2);$ radius: $6$

Ernest C.

Numerade Educator

Center: $(4, -1, 1);$ radius: $5$

Taylor S.

Numerade Educator

Center: $(-3, 7, 5);$ radius: $10$

Ernest C.

Numerade Educator

Center: $(0, 5, -9);$ radius: $8$

Taylor S.

Numerade Educator

Endpoints of a diameter: $(3, 0, 0), (0, 0, 6)$

Ernest C.

Numerade Educator

Endpoints of a diameter: $(1, 0, 0), (0, 5, 0)$

Taylor S.

Numerade Educator

In Exercises 61-70, find the center and radius of the sphere.

$x^2+y^2+z^2-9x=0$

Taylor S.

Numerade Educator

In Exercises 61-70, find the center and radius of the sphere.

$x^2+y^2+z^2-4x+2y-6z+10=0$

Ernest C.

Numerade Educator

In Exercises 61-70, find the center and radius of the sphere.

$x^2+y^2+z^2-6x+4y+9=0$

Taylor S.

Numerade Educator

In Exercises 61-70, find the center and radius of the sphere.

$x^2+y^2+z^2+4x-8z+19=0$

Ernest C.

Numerade Educator

In Exercises 61-70, find the center and radius of the sphere.

$x^2+y^2+z^2-8y-6z+13=0$

Taylor S.

Numerade Educator

In Exercises 61-70, find the center and radius of the sphere.

$9x^2+9y^2+9z^2-18x-6y-72z+73=0$

Ernest C.

Numerade Educator

In Exercises 61-70, find the center and radius of the sphere.

$2x^2+2y^2+2z^2-2x-6y-4z+5=0$

Taylor S.

Numerade Educator

In Exercises 61-70, find the center and radius of the sphere.

$9x^2+9y^2+9z^2-6x+18y+1=0$

Ernest C.

Numerade Educator

In Exercises 61-70, find the center and radius of the sphere.

$4x^2+4y^2+4z^2-4x-32y-8z+33=0$

Taylor S.

Numerade Educator

In Exercises 71-74, sketch the graph of the equation and sketch the specified trace.

$(x-1)^2+y^2+z^2=36; xz$-trace

Ernest C.

Numerade Educator

In Exercises 71-74, sketch the graph of the equation and sketch the specified trace.

$x^2+(y+3)^2+z^2=25; yz$-trace

Taylor S.

Numerade Educator

In Exercises 71-74, sketch the graph of the equation and sketch the specified trace.

$(x+2)^2+(y-3)^2+z^2=9; yz$-trace

Ernest C.

Numerade Educator

In Exercises 71-74, sketch the graph of the equation and sketch the specified trace.

$x^2+(y-1)^2+(z+1)^2=4; xy$-trace

Taylor S.

Numerade Educator

In Exercises 75 and 76, use a three-dimensional graphing utility to graph the sphere.

$x^2+y^2+z^2-6x-8y-10z+46=0$

Ernest C.

Numerade Educator

In Exercises 75 and 76, use a three-dimensional graphing utility to graph the sphere.

$x^2+y^2+z^2+6y-8z+21=0$

Taylor S.

Numerade Educator

ARCHITECTURE A spherical building has a diameter of 205 feet. The center of the building is placed at the origin of a three-dimensional coordinate system. What is the equation of the sphere?

Ernest C.

Numerade Educator

GEOGRAPHY Assume that Earth is a sphere with a radius of 4000 miles. The center of Earth is placed at the origin of a three-dimensional coordinate system.

(a) What is the equation of the sphere?

(b) Lines of longitude that run north-south could be represented by what trace(s)? What shape would each of these traces form?

(c) Lines of latitude that run east-west could be represented by what trace(s)? What shape would each of these traces form?

Taylor S.

Numerade Educator

TRUE OR FALSE? In Exercises 79 and 80, determine whether the statement is true or false. Justify your answer.

In the ordered triple $(x, y, z)$ that represents point $P$ in space, $x$ is the directed distance from the $xy$-plane to $P$.

Ernest C.

Numerade Educator

TRUE OR FALSE? In Exercises 79 and 80, determine whether the statement is true or false. Justify your answer.

The surface consisting of all points $(x, y, z)$ in space that are the same distance $r$ from the point $(h, j, k)$ has a circle as its $xy$-trace.

Taylor S.

Numerade Educator

THINK ABOUT IT What is the $z$-coordinate of any point in the $xy$-plane? What is the $y$-coordinate of any point in the $xz$-plane? What is the $x$-coordinate of any point in the $yz$-plane?

Ernest C.

Numerade Educator

CAPSTONE Find the equation of the sphere that has the points $(3, -2, 6)$ and $(-1, 4, 2)$ as endpoints of a diameter. Explain how this problem gives you a chance to use these formulas: the Distance Formula in Space, the Midpoint Formula in Space, and the standard equation of a sphere.

Taylor S.

Numerade Educator

A line segment has $(x_1, y_1, z_1)$ as one endpoint and $(x_m, y_m, z_m)$ as its midpoint. Find the other endpoint $(x_2, y_2, z_2)$of the line segment in terms of $x_1, y_1, z_1, x_m, y_m,$ and $z_m$.

Ernest C.

Numerade Educator

Use the result of Exercise 85 to find the coordinates of the endpoint of a line segment if the coordinates of the other endpoint and the midpoint are $(3, 0, 2)$ and $(5, 8, 7),$ respectively.

Taylor S.

Numerade Educator