Consider the intersection of the surfaces x² + y² = 1 and x + 2y + z = 0. (a) Sketch the surfaces and the curve of intersection. (b) Find a parametric equation r(t) = (x(t), y(t), z(t)) of the curve of intersection. Hint: Look at the projection of the curve onto the xy-plane, this is a circle which we already know how to parametrize. (c) Find the equation of the tangent line to the curve at the point (0, 1, -2). In other words, compute the tangent vector v of the curve at the point (0, 1, -2) and use the tangent vector to parametrize the line tangent to the curve at (0, 1, -2).
Consider the intersection of the surfaces x² + y² = 1 and x + 2y + z = 0. (a) Sketch the surfaces and the curve of intersection. (b) Find a parametric equation r(t) = (x(t), y(t), z(t)) of the curve of intersection. Hint: Look at the projection of the curve onto the xy-plane, this is a circle which we already know how to parametrize. (c) Find the equation of the tangent line to the curve at the point (0, 1, -2). In other words, compute the tangent vector v of the curve at the point (0, 1, -2) and use the tangent vector to parametrize the line tangent to the curve at (0, 1, -2).
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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Transcribed Image Text:Consider the intersection of the surfaces x² + y² = 1 and x + 2y + z = 0.
(a)
Sketch the surfaces and the curve of intersection.
(b)
Find a parametric equation r(t) = (x(t), y(t), z(t)) of the curve of intersection.
Hint: Look at the projection of the curve onto the xy-plane, this is a circle which we
already know how to parametrize.
(c)
Find the equation of the tangent line to the curve at the point (0, 1, -2). In other
words, compute the tangent vector v of the curve at the point (0, 1, -2) and use the
tangent vector to parametrize the line tangent to the curve at (0, 1, -2).
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