Stewart Precalc 6e Section 9.3: Three-Dimensional Coordinate Geometry

1 Concept - 3D Coordinate System · Level 1

Refer to the figure. In a three-dimensional coordinate system the three mutually perpendicular axes are called the ___-axis, the ___-axis, and the ___-axis. Label the axes in the figure. The point \(P\) in the figure has coordinates (___, ___, ___). The equation of the plane passing through \(P\) and parallel to the \(x z\)-plane is ___.

2 Concept - Distance and Sphere · Level 1

Refer to the figure. The distance between the point \(P(x_1, y_1, z_1)\) and \(Q(x_2, y_2, z_2)\) is given by the formula \(d(P, Q) = \) ___. The distance between the point \(P\) in the figure and the origin is ___. The equation of the sphere centered at \(P\) with radius 3 is ___.

3 Skill - Distance Between Points · Level 2

Two points \(P\) and \(Q\) are given. (a) Plot \(P\) and \(Q\). (b) Find the distance between \(P\) and \(Q\). \(P(3, 1, 0)\), \(Q(-1, 2, -5)\)

4 Skill - Distance Between Points · Level 2

Two points \(P\) and \(Q\) are given. (a) Plot \(P\) and \(Q\). (b) Find the distance between \(P\) and \(Q\). \(P(5, 0, 10)\), \(Q(3, -6, 7)\)

5 Skill - Distance Between Points · Level 2

Two points \(P\) and \(Q\) are given. (a) Plot \(P\) and \(Q\). (b) Find the distance between \(P\) and \(Q\). \(P(-2, -1, 0)\), \(Q(-12, 3, 0)\)

6 Skill - Distance Between Points · Level 2

Two points \(P\) and \(Q\) are given. (a) Plot \(P\) and \(Q\). (b) Find the distance between \(P\) and \(Q\). \(P(5, -4, -6)\), \(Q(8, -7, 4)\)

7 Skill - Surfaces in 3D · Level 2

Describe and sketch the surface represented by the given equation. \(x = 4\)

8 Skill - Surfaces in 3D · Level 2

Describe and sketch the surface represented by the given equation. \(y = -2\)

9 Skill - Surfaces in 3D · Level 2

Describe and sketch the surface represented by the given equation. \(z = 8\)

10 Skill - Surfaces in 3D · Level 2

Describe and sketch the surface represented by the given equation. \(y = -1\)

11 Skill - Equation of a Sphere · Level 2

Find an equation of a sphere with the given radius \(r\) and center \(C\). \(r = 5\); \(C(2, -5, 3)\)

12 Skill - Equation of a Sphere · Level 2

Find an equation of a sphere with the given radius \(r\) and center \(C\). \(r = 3\); \(C(-1, 4, -7)\)

13 Skill - Equation of a Sphere · Level 2

Find an equation of a sphere with the given radius \(r\) and center \(C\). \(r = \sqrt{6}\); \(C(3, -1, 0)\)

14 Skill - Equation of a Sphere · Level 2

Find an equation of a sphere with the given radius \(r\) and center \(C\). \(r = \sqrt{11}\); \(C(-10, 0, 1)\)

15 Skill - Center and Radius of a Sphere · Level 3

Show that the equation represents a sphere, and find its center and radius. \(x^2 + y^2 + z^2 - 10x + 2y + 8z = 9\)

16 Skill - Center and Radius of a Sphere · Level 3

Show that the equation represents a sphere, and find its center and radius. \(x^2 + y^2 + z^2 + 4x - 6y + 2z = 10\)

17 Skill - Center and Radius of a Sphere · Level 3

Show that the equation represents a sphere, and find its center and radius. \(x^2 + y^2 + z^2 = 12x + 2y\)

18 Skill - Center and Radius of a Sphere · Level 3

Show that the equation represents a sphere, and find its center and radius. \(x^2 + y^2 + z^2 = 14y - 6z\)

19 Skill - Trace of a Sphere · Level 3

Describe the trace of the sphere \((x + 1)^2 + (y - 2)^2 + (z + 10)^2 = 100\) in (a) the \(y z\)-plane and (b) the plane \(x = 4\).

20 Skill - Trace of a Sphere · Level 3

Describe the trace of the sphere \(x^2 + (y - 4)^2 + (z - 3)^2 = 144\) in (a) the \(x z\)-plane and (b) the plane \(z = -2\).

21 Application - Spherical Water Tank · Level 3

Spherical Water Tank. A water tank is in the shape of a sphere of radius 5 feet. The tank is supported on a metal circle 4 feet below the center of the sphere, as shown in the figure. Find the radius of the metal circle.

22 Application - Spherical Buoy · Level 3

A Spherical Buoy. A spherical buoy of radius 2 feet floats in a calm lake. Six inches of the buoy are submerged. Place a coordinate system with the origin at the center of the sphere.

(a) Find an equation of the sphere.

Answer for 22(a)

(b) Find an equation of the circle formed at the waterline of the buoy.

Answer for 22(b)

Enter your answer directly below each part above.

23 Application - Visualizing a Set in Space · Level 4

Visualizing a Set in Space. Try to visualize the set of all points \((x, y, z)\) in a coordinate space that are equidistant from the points \(P(0, 0, 0)\) and \(Q(0, 3, 0)\). Use the Distance Formula to find an equation for this surface, and observe that it is a plane.

24 Application - Visualizing a Set in Space · Level 4

Visualizing a Set in Space. Try to visualize the set of all points \((x, y, z)\) in a coordinate space that are twice as far from the point \(Q(0, 3, 0)\) as from the point \(P(0, 0, 0)\). Use the Distance Formula to show that the set is a sphere, and find its center and radius.

25 Example - Plotting Points in Three Dimensions · Level 1

Plot the points \((2, 4, 7)\) and \((-4, 3, -5)\) in the three-dimensional rectangular coordinate system.

26 Example - Sketching Coordinate Planes · Level 2

Describe and sketch the surface in three-dimensional space represented by the equation.

(a) \(z = 3\)

Answer for 26(a)

(b) \(y = 5\)

Answer for 26(b)

Enter your answer directly below each part above.

27 Example - Distance Formula in 3D · Level 2

Find the distance between the points \(P(2, -1, 7)\) and \(Q(1, -3, 5)\).

28 Example - Equation of a Sphere · Level 2

Find an equation of a sphere with radius 5 and center \(C(-2, 1, 3)\).

29 Example - Center and Radius of a Sphere · Level 3

Show that \(x^2 + y^2 + z^2 + 4x - 6y + 2z + 6 = 0\) is the equation of a sphere, and find its center and radius.

30 Example - Trace of a Sphere · Level 3

Describe the trace of the sphere \((x - 2)^2 + (y - 4)^2 + (z - 5)^2 = 36\) in (a) the \(x y\)-plane and (b) the plane \(z = 9\).

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