a) If u and v are two orthogonal vectors with ||u|| = ||v|| and ro (x0, Y0, z0), show that r(t) : = ro + (cos t)u + (sin t)v, 0 1, show that a(a – 1) a(a – 1) cos t, V a ra(t) sin t, V a + sin 2t (cost + sin t) %3D a + sin 2t a + sin 2t parametrizes a circle centered at the origin. What is the radius of this circle, and what is an equation for the plane this circle lies in?

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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(a) If u and v are two orthogonal vectors with ||u|| = ||v|| and ro = (x0, Y0, zo), show that
r(t) = ro + (cos t)u+ (sin t) v,
0<t< 2n
parametrizes a circle of radius ||u|| = ||v|| and center (xo, Yo, zo). (Your work will be much
cleaner if you use properties of dot products like ||x||2
components.)
= x•x instead of trying to use
(b) While the parametrization from (a) is the simplest way to parametrize an arbitrary circle,
there are of course many more complicated possibilities. For a > 1, show that
a (α-1)
a(а — 1)
sin t,
a + sin 2t
a
ra(t) =
cos t,
a + sin 2t
(cost+ sin t)
a + sin 2t
parametrizes a circle centered at the origin. What is the
an equation for the plane this circle lies in?
of this circle, and
hat is
Transcribed Image Text:(a) If u and v are two orthogonal vectors with ||u|| = ||v|| and ro = (x0, Y0, zo), show that r(t) = ro + (cos t)u+ (sin t) v, 0<t< 2n parametrizes a circle of radius ||u|| = ||v|| and center (xo, Yo, zo). (Your work will be much cleaner if you use properties of dot products like ||x||2 components.) = x•x instead of trying to use (b) While the parametrization from (a) is the simplest way to parametrize an arbitrary circle, there are of course many more complicated possibilities. For a > 1, show that a (α-1) a(а — 1) sin t, a + sin 2t a ra(t) = cos t, a + sin 2t (cost+ sin t) a + sin 2t parametrizes a circle centered at the origin. What is the an equation for the plane this circle lies in? of this circle, and hat is
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