Section 1
The Three-Dimensional Coordinate System
In Exercises $1-4,$ plot the points on the same three dimensional coordinate system.$$\begin{array}{l}{\text { (a) }(2,1,3)} \\ {\text { (b) }(-1,2,1)}\end{array}$$
Plot the points on the same three dimensional coordinate system.$$\begin{array}{l}{\text { (a) }(3,-2,5)} \\ {\text { (b) }\left(\frac{3}{2}, 4,-2\right)}\end{array}$$
Plot the points on the same three dimensional coordinate system.$$\begin{array}{l}{\text { (a) }(5,-2,2)} \\ {\text { (b) }(5,-2,-2)}\end{array}$$
Plot the points on the same three dimensional coordinate system.$$\begin{array}{l}{\text { (a) }(0,4,-5)} \\ {\text { (b) }(4,0,5)}\end{array}$$
In Exercises 5 and $6,$ approximate the coordinates of the points.(Graph cant copy)
In Exercises $7-10,$ find the coordinates of the point.$$\begin{array}{l}{\text { The point is located three units behind the } y z \text { -plane, four }} \\ {\text { units to the right of the } x z \text { -plane, and five units above the }} \\ {x y \text { -plane. }}\end{array}$$
Find the coordinates of the point.$$\begin{array}{l}{\text { The point is located seven units in front of the } y z \text { -plane, two }} \\ {\text { units to the left of the } x z \text { -plane, and one unit below the }} \\ {x y \text { -plane. }}\end{array}$$
Find the coordinates of the point.$$\begin{array}{l}{\text { The point is located on the } x \text { -axis, } 10 \text { units in front of the }} \\ {y z \text { -plane. }}\end{array}$$
Find the coordinates of the point.$$\begin{array}{l}{\text { The point is located in the } y z \text { -plane, three units to the right }} \\ {\text { of the } x z \text { -plane, and two units above the } x y \text { -plane. }}\end{array}$$
Think About It What is the $z$ -coordinate of any point in the $x$ -plane?
Think About It What is the $x$ -coordinate of any point in the $y z$ -plane?
In Exercises $13-16,$ find the distance between the two points.$$(4,1,5),(8,2,6)$$
Find the distance between the two points.$$(-4,-1,1),(2,-1,5)$$
Find the distance between the two points.$$(-1,-5,7),(-3,4,-4)$$
Find the distance between the two points.$$(8,-2,2),(8,-2,4)$$
In Exercises $17-20,$ find the coordinates of the midpoint of the line segment joining the two points.$$(6,-9,1),(-2,-1,5)$$
Find the coordinates of the midpoint of the line segment joining the two points.$$(4,0,-6),(8,8,20)$$
Find the coordinates of the midpoint of the line segment joining the two points.$$(-5,-2,5),(6,3,-7)$$
Find the coordinates of the midpoint of the line segment joining the two points.$$(0,-2,5),(4,2,7)$$
In Exercises $21-24,$ find $(x, y, z)$(Graph cant copy)
Find $(x, y, z)$(Graph cant copy)
In Exercises $25-28,$ find the lengths of the sides of the triangle with the given vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither of these.$$(0,0,0),(2,2,1),(2,-4,4)$$
Find the lengths of the sides of the triangle with the given vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither of these.$$(5,3,4),(7,1,3),(3,5,3)$$
Find the lengths of the sides of the triangle with the given vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither of these.$$(-2,2,4),(-2,2,6),(-2,4,8)$$
Find the lengths of the sides of the triangle with the given vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither of these.$$(5,0,0),(0,2,0),(0,0,-3)$$
Think About It The triangle in Exercise 25 is translated five units upward along the $z$ -axis. Determine the coordinates of the translated triangle.
Think About It The triangle in Exercise 26 is translated three units to the right along the $y$ -axis. Determine the coordinates of the translated triangle.
In Exercises $31-40,$ find the standard equation of the sphere.(Sphere cant copy)
Find the standard equation of the sphere.(Sphere cant copy)
Find the standard equation of the sphere.$$\text { Center: }(1,1,5) ; \text { radius: } 3$$
Find the standard equation of the sphere.$$\text { Center: }(4,-1,1) ; \text { radius: } 5$$
Find the standard equation of the sphere.$$\text { Endpoints of a diameter: }(2,0,0),(0,6,0)$$
Find the standard equation of the sphere.$$\text { Endpoints of a diameter: }(1,0,0),(0,5,0)$$
Find the standard equation of the sphere.$$\text { Center: }(-2,1,1) ; \text { tangent to the } x y \text { -plane }$$
Find the standard equation of the sphere.$$\text { Center: }(1,2,0) ; \text { tangent to the } y z \text { -plane }$$
In Exercises $41-46,$ find the sphere's center and radius.$$x^{2}+y^{2}+z^{2}-5 x=0$$
Find the sphere's center and radius.$$x^{2}+y^{2}+z^{2}-8 y=0$$
Find the sphere's center and radius.$$x^{2}+y^{2}+z^{2}-2 x+6 y+8 z+1=0$$
Find the sphere's center and radius.$$x^{2}+y^{2}+z^{2}-4 y+6 z+4=0$$
Find the sphere's center and radius.$$2 x^{2}+2 y^{2}+2 z^{2}-4 x-12 y-8 z+3=0$$
Find the sphere's center and radius.$$4 x^{2}+4 y^{2}+4 z^{2}-8 x+16 y+11=0$$
In Exercises $47-50,$ sketch the $x y$ -trace of the sphere.$$(x-1)^{2}+(y-3)^{2}+(z-2)^{2}=25$$
Sketch the $x y$ -trace of the sphere.$$(x+1)^{2}+(y+2)^{2}+(z-2)^{2}=16$$
Sketch the $x y$ -trace of the sphere.$$x^{2}+y^{2}+z^{2}-6 x-10 y+6 z+30=0$$
Sketch the $x y$ -trace of the sphere.$$x^{2}+y^{2}+z^{2}-4 y+2 z-60=0$$
In Exercises $51-54,$ sketch the $y z$ -trace of the sphere.$$x^{2}+(y+3)^{2}+z^{2}=25$$
Sketch the $y z$ -trace of the sphere.$$(x+2)^{2}+(y-3)^{2}+z^{2}=9$$
Sketch the $y z$ -trace of the sphere.$$x^{2}+y^{2}+z^{2}-4 x-4 y-6 z-12=0$$
Sketch the $y z$ -trace of the sphere.$$x^{2}+y^{2}+z^{2}-6 x-10 y+6 z+30=0$$
In Exercises $55-58,$ sketch the trace of the intersection of each plane with the given sphere.
$$\begin{array}{l}{x^{2}+y^{2}+z^{2}=25} \\ {\text { (a) } z=3 \quad \text { (b) } x=4}\end{array}$$
Sketch the trace of the intersection of each plane with the given sphere.$$\begin{array}{l}{x^{2}+y^{2}+z^{2}=169} \\ {\text { (a) } x=5 \quad \text { (b) } y=12}\end{array}$$
Sketch the trace of the intersection of each plane with the given sphere.$$\begin{array}{l}{x^{2}+y^{2}+z^{2}-4 x-6 y+9=0} \\ {\text { (a) } x=2 \quad \text { (b) } y=3}\end{array}$$
Sketch the trace of the intersection of each plane with the given sphere.$$\begin{array}{l}{x^{2}+y^{2}+z^{2}-8 x-6 z+16=0} \\ {\text { (a) } x=4 \quad \text { (b) } z=3}\end{array}$$
Geology Crystals are classified according to their symmetry. Crystals shaped like cubes are classified as isometric. The vertices of an isometric crystal mapped onto a three-dimensional coordinate system are shown in the figure. Determine $(x, y, z) .$(Figure cant copy)
Crystals Crystals shaped like rectangular prisms are classified as tetragonal. The vertices of a tetragonal crystal mapped onto a three-dimensional coordinate system are shown in the figure. Determine $(x, y, z) .$(Figure cant copy)
Architecture A spherical building has a diameter of 165 feet. The center of the building is placed at the origin of a three-dimensional coordinate system. What is the equation of the sphere?