5. Betweenness in taxicab geometry. Let A (0,0), B = (1,0) and C = (1, 1) and let p denote the taxicab metric. (a) Find all points P such that p(A, P) + p(P, B) = p(A, B). Draw a sketch in the Cartesian plane. (b) Find all points P such that p(A, P) + p(P, C) = p(A, C). Draw a sketch in the Cartesian plane.
5. Betweenness in taxicab geometry. Let A (0,0), B = (1,0) and C = (1, 1) and let p denote the taxicab metric. (a) Find all points P such that p(A, P) + p(P, B) = p(A, B). Draw a sketch in the Cartesian plane. (b) Find all points P such that p(A, P) + p(P, C) = p(A, C). Draw a sketch in the Cartesian plane.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![FIGURE 3.7: Two circles in the rational plane
coordinate functions use real numbers will allow us to fill that gap. Specifically, we will
prove that if one circle contains a point that is inside a second circle and another point
that is outside the second circle, then the two circles must intersect. Even using the
powerful axioms in this chapter the proof is not easy, however, and we will not attempt
that particular proof until Chapter 8.
EXERCISES 3.2
1. Prove: If l and m are two lines, the number of points in l n m is either 0, 1, or oo.
2. Show that the Euclidean metric defined in Example 3.2.10 is a metric (i.e., verify that
the function d satisfies the three conditions in the definition of metric on page 39).
3. Show that the taxicab metric defined in Example 3.2.11 is a metric (i.e., verify that the
function p satisfies the three conditions in the definition of metric on page 39).
4. Show that the spherical metric defined in Example 3.2.12 is a metric (i.e., verify that the
function s satisfies the three conditions in the definition of metric on page 39).
5. Betweenness in taxicab geometry. Let A =
denote the taxicab metric.
(0, 0), В 3D (1, 0) and C 3D
(1,1) and let p
(a) Find all points P such that p( A, P) + p(P, B) = p(A, B). Draw a sketch in the
Cartesian plane.
(b) Find all points P such that p(A, P) + p(P, C) = p(A, C). Draw a sketch in the
Cartesian plane.
6. Betweenness on the sphere. Let A, B, and C be points on the sphere S2. Define C to be
between A and B if A, B, and C are collinear and s(A, C) + s(C, B) = s(A, B), where
s is the metric defined in Example 3.2.12. Also define segment in the usual way.
(a) Find all points that are between A and C in case A and C are nonantipodal points.
Sketch the segment from A to C.
(b) Find all points that are between A and C in case A and C are antipodal points.
Sketch the segment from A to C.
7. Find all points (x, y) in R² such that p((0,0), (x, y))
Draw a sketch in the Cartesian plane. (This shape might be called a "circle" in the
taxicab metric.)
8. The square metric. Define the distance between two points (x1, yı) and (x2, y2) in R²
by D( (x1, yı), (x2, y2 )) = max{\x2
(a) Verify that D is a metric.
(b) Find all points (x, y) in R? such that D((0,0), (x, y))
Cartesian plane. (This should explain the name square metric.)
1, where p is the taxicab metric.
x1\, \y2
yıl}.4
1. Draw a sketch in the
4max{a, b} denotes the larger of the two real numbers a and b.
46
Chapter 3
Axioms for Plane Geometry
9. Verify that the functions defined in Example 3.2.14 are coordinate functions.
10. Verify that the functions defined in Example 3.2.15 are coordinate functions.
11. Let l be a line in the Cartesian plane R². Find a function f : l → R that is a coordinate
function for l in the square metric. (See Exercise 8 for definition of square metric.)
12. Assume that f : l → R is a coordinate function for l.
(a) Prove that – f is also a coordinate function for l.
(b) Prove that g : l → R defined by g(P) = f(P) + c for some constant c is also a
coordinate function for l.
(c) Prove that if h : l → R is any coordinate function for l then there must exist a
constant c such that either h(P) = f(P) + c or h( P) = –f(P) + c.
13. Let l be a line and let f : l → R be a function such that PQ = \f(P) – ƒ(Q)| for
every P, Q E l. Prove that f is a coordinate function for l.
14. Prove the following fact from high school algebra that was needed in the proof of
Theorem 3.2.17: If x and y are two nonzero real numbers such that |x| + ]y| = ]x + yl,
then either both x and y are positive or both x and y are negative.
15. Prove Corollary 3.2.20.
16. Prove that if C E AB and C # A, then AŔ = AĆ.
17. Prove existence and uniqueness of midpoints (Theorem 3.2.22).
18. (Segment Construction Theorem) Prove the following theorem. If AB is a segment and
CD is a ray, then there is a unique point E on CD such that AB = CE.
19. (Segment Addition Theorem) Prove the following theorem. If A * B * C, D * E * F,
AB = DE, and BC = EF, then AC = DF.
20. (Segment Subtraction Theorem) Prove the following theorem. If A * B * C, D * E * F,
AB = DE, and AC = DF, then BC = EF.
21. Let A and B be two distinct points. Prove that AB
22. Prove that the endpoints of a segment are well defined (i.e., if AB
A = C and B = D or A = D and B =
ВА.
CD, then either
C).
23. Let A and B be two distinct points. Prove each of the following equalities.
2. . 3?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadf70025-15e9-4d93-968e-9d86049b61bd%2F5ae4db8b-aaaf-4818-98e8-a51f84cf650c%2Fwq18a5w_processed.png&w=3840&q=75)
Transcribed Image Text:FIGURE 3.7: Two circles in the rational plane
coordinate functions use real numbers will allow us to fill that gap. Specifically, we will
prove that if one circle contains a point that is inside a second circle and another point
that is outside the second circle, then the two circles must intersect. Even using the
powerful axioms in this chapter the proof is not easy, however, and we will not attempt
that particular proof until Chapter 8.
EXERCISES 3.2
1. Prove: If l and m are two lines, the number of points in l n m is either 0, 1, or oo.
2. Show that the Euclidean metric defined in Example 3.2.10 is a metric (i.e., verify that
the function d satisfies the three conditions in the definition of metric on page 39).
3. Show that the taxicab metric defined in Example 3.2.11 is a metric (i.e., verify that the
function p satisfies the three conditions in the definition of metric on page 39).
4. Show that the spherical metric defined in Example 3.2.12 is a metric (i.e., verify that the
function s satisfies the three conditions in the definition of metric on page 39).
5. Betweenness in taxicab geometry. Let A =
denote the taxicab metric.
(0, 0), В 3D (1, 0) and C 3D
(1,1) and let p
(a) Find all points P such that p( A, P) + p(P, B) = p(A, B). Draw a sketch in the
Cartesian plane.
(b) Find all points P such that p(A, P) + p(P, C) = p(A, C). Draw a sketch in the
Cartesian plane.
6. Betweenness on the sphere. Let A, B, and C be points on the sphere S2. Define C to be
between A and B if A, B, and C are collinear and s(A, C) + s(C, B) = s(A, B), where
s is the metric defined in Example 3.2.12. Also define segment in the usual way.
(a) Find all points that are between A and C in case A and C are nonantipodal points.
Sketch the segment from A to C.
(b) Find all points that are between A and C in case A and C are antipodal points.
Sketch the segment from A to C.
7. Find all points (x, y) in R² such that p((0,0), (x, y))
Draw a sketch in the Cartesian plane. (This shape might be called a "circle" in the
taxicab metric.)
8. The square metric. Define the distance between two points (x1, yı) and (x2, y2) in R²
by D( (x1, yı), (x2, y2 )) = max{\x2
(a) Verify that D is a metric.
(b) Find all points (x, y) in R? such that D((0,0), (x, y))
Cartesian plane. (This should explain the name square metric.)
1, where p is the taxicab metric.
x1\, \y2
yıl}.4
1. Draw a sketch in the
4max{a, b} denotes the larger of the two real numbers a and b.
46
Chapter 3
Axioms for Plane Geometry
9. Verify that the functions defined in Example 3.2.14 are coordinate functions.
10. Verify that the functions defined in Example 3.2.15 are coordinate functions.
11. Let l be a line in the Cartesian plane R². Find a function f : l → R that is a coordinate
function for l in the square metric. (See Exercise 8 for definition of square metric.)
12. Assume that f : l → R is a coordinate function for l.
(a) Prove that – f is also a coordinate function for l.
(b) Prove that g : l → R defined by g(P) = f(P) + c for some constant c is also a
coordinate function for l.
(c) Prove that if h : l → R is any coordinate function for l then there must exist a
constant c such that either h(P) = f(P) + c or h( P) = –f(P) + c.
13. Let l be a line and let f : l → R be a function such that PQ = \f(P) – ƒ(Q)| for
every P, Q E l. Prove that f is a coordinate function for l.
14. Prove the following fact from high school algebra that was needed in the proof of
Theorem 3.2.17: If x and y are two nonzero real numbers such that |x| + ]y| = ]x + yl,
then either both x and y are positive or both x and y are negative.
15. Prove Corollary 3.2.20.
16. Prove that if C E AB and C # A, then AŔ = AĆ.
17. Prove existence and uniqueness of midpoints (Theorem 3.2.22).
18. (Segment Construction Theorem) Prove the following theorem. If AB is a segment and
CD is a ray, then there is a unique point E on CD such that AB = CE.
19. (Segment Addition Theorem) Prove the following theorem. If A * B * C, D * E * F,
AB = DE, and BC = EF, then AC = DF.
20. (Segment Subtraction Theorem) Prove the following theorem. If A * B * C, D * E * F,
AB = DE, and AC = DF, then BC = EF.
21. Let A and B be two distinct points. Prove that AB
22. Prove that the endpoints of a segment are well defined (i.e., if AB
A = C and B = D or A = D and B =
ВА.
CD, then either
C).
23. Let A and B be two distinct points. Prove each of the following equalities.
2. . 3?
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