Consider the following curves: r₁(t) = (sint, sint, √2 cost), r2(t)=(√2 cost, √2 sint,0), t = [0, 2π] (a) Find the points where the curves intersect (b) Find the angle between these curves at the points of intersection (c) Show that the curves are circles on a sphere. Answer: (a) The curves intersect at (1,1,0) and (-1,-1,0). (b) the curves are orthogonal at these points.
Consider the following curves: r₁(t) = (sint, sint, √2 cost), r2(t)=(√2 cost, √2 sint,0), t = [0, 2π] (a) Find the points where the curves intersect (b) Find the angle between these curves at the points of intersection (c) Show that the curves are circles on a sphere. Answer: (a) The curves intersect at (1,1,0) and (-1,-1,0). (b) the curves are orthogonal at these points.
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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![3. Consider the following curves:
r₁(t) = (sint, sint, √2 cost), r2(t)=(√2 cost, √2 sint, 0), t = [0, 2π]
(a) Find the points where the curves intersect
(b) Find the angle between these curves at the points of intersection
(c) Show that the curves are circles on a sphere.
Answer: (a) The curves intersect at (1,1,0) and (-1,-1,0). (b) the curves are orthogonal at these points.
4. Show that the curve ri(t) = (et, et sint, et cos t) is on a cone. Find an equation for the tangent
line at t = 0.
Answer: The curve is on the cone r² = y² + 2²; z = 1+t₁y = t₁ z = 1+ t
5. Show that all points P = (x, y, z) that are equidistant from point Q = (0, -1,0)and the plane
y = 1 lie on a quadric surface. Find the equation of the quadric, identify this surface and sketch
it.
Answer: The curve is on the elliptic paraboloid 2² +2² -4y
6. Find the spherical coordinates of a point with cylindrical coordinates (1/√2, π/4, 1/√2). What
is the surface of spherical equation p= 2 cos + 4 sino sin 0 ?
Answer: (1, n/4, /4); a sphere centered at (0, 2, 1) of radius √5.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F640887de-34e2-4fb5-a217-028aef2f5eb9%2F1a2c8761-504c-46ff-9a1c-2c939668a775%2Fvk7s3bb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Consider the following curves:
r₁(t) = (sint, sint, √2 cost), r2(t)=(√2 cost, √2 sint, 0), t = [0, 2π]
(a) Find the points where the curves intersect
(b) Find the angle between these curves at the points of intersection
(c) Show that the curves are circles on a sphere.
Answer: (a) The curves intersect at (1,1,0) and (-1,-1,0). (b) the curves are orthogonal at these points.
4. Show that the curve ri(t) = (et, et sint, et cos t) is on a cone. Find an equation for the tangent
line at t = 0.
Answer: The curve is on the cone r² = y² + 2²; z = 1+t₁y = t₁ z = 1+ t
5. Show that all points P = (x, y, z) that are equidistant from point Q = (0, -1,0)and the plane
y = 1 lie on a quadric surface. Find the equation of the quadric, identify this surface and sketch
it.
Answer: The curve is on the elliptic paraboloid 2² +2² -4y
6. Find the spherical coordinates of a point with cylindrical coordinates (1/√2, π/4, 1/√2). What
is the surface of spherical equation p= 2 cos + 4 sino sin 0 ?
Answer: (1, n/4, /4); a sphere centered at (0, 2, 1) of radius √5.
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