a,b,&c 7. In this problem, we prove that a straight line is the shortest curve between two points inR. 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.to1(c) Show that the arc length of r from r(to) to r(t1) is at least ||q- p||. Hint: Considera well-chosen unit vector u.

Answered: 7. In this problem, we prove that a…

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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Related questions

Question

a,b,&c

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

Given - $l e t p, q \in ℝ^{d} a n d r b e a c u r v e s u c h t h a t r (t_{0}) = p a n d r (t_{1}) = q w h e r e t_{0} < t_{1}$

(a)

let r be a curve passing through point p and q

$T o s h o w - i f u i s a u n i t v e c t o r i n ℝ^{d}, t h e r' (t) . u \leq ∥ r' (t) ∥ f o r a l l t$

$a s w e k n o w r : [t_{0}, t_{1}] \to ℝ^{d} b e d e f i n e d b y \Rightarrow r (t) = [r_{1} (t) r_{2} (t) . . . . . . . . . . . r_{d} (t)] w h e r e t \in [t_{0}, t_{1}] r : [t_{0}, t_{1}] \to ℝ^{d} i s a r e a l v a l u e \Rightarrow ∥ r (t) ∥ = \sqrt{{[r_{1} (t)]}^{2} + [r_{2} {(t)}^{2} + . . . . . . . . . . + {[r_{d} (t)]}^{2}}$