Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
7. In this problem, we prove that a straight line is the shortest curve between two points in Rd. Let p, q E Rd and let r be a curve such that r(to) = p and r(tı) = q, where to < t1. (a) Show that, if u is any unit vector in Rd, the r'(t) · u < ||r'(t)|| for all t. (b) Show that t1 (q - p) · u< / ||r(t)||dt. 1

7. In this problem, we prove that a straight line is the shortest curve between two points in Rd. Let p, q E Rd and let r be a curve such that r(to) = p and r(tı) = q, where to < t1. (a) Show that, if u is any unit vector in Rd, the r'(t) · u < ||r'(t)|| for all t. (b) Show that t1 (q - p) · u< / ||r(t)||dt. 1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
7. In this problem, we prove that a straight line is the shortest curve between two points in R. Let p, q E Rª and let r be a curve such that r(to) = p and r(t1) = q, where to < t1. (a) Show that, if u is any unit vector in Rd, the r'(t) · u < ||r'(t)|| for all t. • u (b) Show that (q – p) · u < / ||r (t)||dt. to 1 (c) Show that the arc length of r from r(to) to r(t1) is at least ||q- p||. Hint: Consider a well-chosen unit vector u.
Transcribed Image Text:7. In this problem, we prove that a straight line is the shortest curve between two points in R. Let p, q E Rª and let r be a curve such that r(to) = p and r(t1) = q, where to < t1. (a) Show that, if u is any unit vector in Rd, the r'(t) · u < ||r'(t)|| for all t. • u (b) Show that (q – p) · u < / ||r (t)||dt. to 1 (c) Show that the arc length of r from r(to) to r(t1) is at least ||q- p||. Hint: Consider a well-chosen unit vector u.
Expert Solution
Step 1

let p,q∈ℝd and let r be a curve such that rt0=p and rt1=q where t0<t1

(a) show that, if u is any unit vector in ℝd

r't·u≤r't for all t

(b) q-p·u≤∫t0t1r't dt

trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS