Determine whether the following statements are true and give an explanation or counter example. a. Any point on the z-axis has more than one representation in both cylindrical and spherical coordinates. b. The sets {(r,0,z): r=z} and {(p,q,0): q=π/4} are the same. system. B. The statement is false. The origin (x,y,z) = (0,0,0) has only one unique representation in spherical coordinates. C. The statement is false. The origin (x,y,z) = (0,0,0) has only one unique representation in cylindrical coordinates. D. The statement is true. In cylindrical coordinates, any point (x,y,z) = (0,0,z) on the z-axis is represented by (r,0,z)=(0,0,z), where 0 is arbitrary. In spherical coordinates, the point (x,y,z) = (0,0,z) can be represented by (p,q,0) (2,0,0) or (p,q,0) = (-Z,л,0) where 0 is arbitrary. b. Choose the correct answer below. A. The statement is false. The set {(p,q,0): Q =π/4} describes a conical surface and the set {(r,0,z): r = z} describes a circular cylinder. B. The statement is false. The set {(p,q,0): op=π/4} includes a conical surface centered along the negative z-axis and {(r,e,z): r = z} does not. C. The statement is false. The set {(r,0,z): r = z} includes a conical surface centered along the negative z-axis and {(p,q,0): pπ/4} does not. D. The statement is true. Both sets describe the same conical surface with the vertex at the origin and extending in the positive z-direction.
Determine whether the following statements are true and give an explanation or counter example. a. Any point on the z-axis has more than one representation in both cylindrical and spherical coordinates. b. The sets {(r,0,z): r=z} and {(p,q,0): q=π/4} are the same. system. B. The statement is false. The origin (x,y,z) = (0,0,0) has only one unique representation in spherical coordinates. C. The statement is false. The origin (x,y,z) = (0,0,0) has only one unique representation in cylindrical coordinates. D. The statement is true. In cylindrical coordinates, any point (x,y,z) = (0,0,z) on the z-axis is represented by (r,0,z)=(0,0,z), where 0 is arbitrary. In spherical coordinates, the point (x,y,z) = (0,0,z) can be represented by (p,q,0) (2,0,0) or (p,q,0) = (-Z,л,0) where 0 is arbitrary. b. Choose the correct answer below. A. The statement is false. The set {(p,q,0): Q =π/4} describes a conical surface and the set {(r,0,z): r = z} describes a circular cylinder. B. The statement is false. The set {(p,q,0): op=π/4} includes a conical surface centered along the negative z-axis and {(r,e,z): r = z} does not. C. The statement is false. The set {(r,0,z): r = z} includes a conical surface centered along the negative z-axis and {(p,q,0): pπ/4} does not. D. The statement is true. Both sets describe the same conical surface with the vertex at the origin and extending in the positive z-direction.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.4: Plane Curves And Parametric Equations
Problem 38E
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Transcribed Image Text:Determine whether the following statements are true and give an explanation or counter example.
a. Any point on the z-axis has more than one representation in both cylindrical and spherical coordinates.
b. The sets {(r,e,z): r=z} and {(p,q,0): Q =π/4} are the same.
O
system.
B. The statement is false. The origin (x,y,z) = (0,0,0) has only one unique representation in spherical
coordinates.
C. The statement is false. The origin (x,y,z) = (0,0,0) has only one unique representation in cylindrical
coordinates.
'D. The statement is true. In cylindrical coordinates, any point (x,y,z) = (0,0,z) on the z-axis is represented by
(r,0,z)=(0,0,z), where 0 is arbitrary. In spherical coordinates, the point (x,y,z) = (0,0,z) can be represented by
(p,q,0)=(2,0,0) or (p,q,0) = (-Z,л,0) where 0 is arbitrary.
b. Choose the correct answer below.
○ A. The statement is false. The set {(p,q,0): Q =π/4} describes a conical surface and the set {(r,0,z): r = z}
describes a circular cylinder.
B. The statement is false. The set {(p,q,0): =π/4} includes a conical surface centered along the negative
z-axis and {(r,e,z): r = z} does not.
C. The statement is false. The set {(r,0,z): r = z} includes a conical surface centered along the negative z-axis
and {(p,q,0): π/4} does not.
D. The statement is true. Both sets describe the same conical surface with the vertex at the origin and
extending in the positive z-direction.
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