(a) Suppose that the intersection of two surfaces F(x, y, z) = 0 and G(x, y, z) = 0 is a curve C. Explain why the vector v = VFp × VGp is a direction vector for the tangent line to C at P. (b) Let C be the curve of intersection of the spheres x² + y² + z² = 3 and (x − 2)² + (y − 2)² + z² = 3. Use the result of part (a) to find parameric equations of the tangent line to C at P = (1, 1, 1).
(a) Suppose that the intersection of two surfaces F(x, y, z) = 0 and G(x, y, z) = 0 is a curve C. Explain why the vector v = VFp × VGp is a direction vector for the tangent line to C at P. (b) Let C be the curve of intersection of the spheres x² + y² + z² = 3 and (x − 2)² + (y − 2)² + z² = 3. Use the result of part (a) to find parameric equations of the tangent line to C at P = (1, 1, 1).
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:(a) Suppose that the intersection of two surfaces \( F(x,y,z) = 0 \) and \( G(x,y,z) = 0 \) is a curve \( C \). Explain why the vector \( \mathbf{v} = \nabla F_P \times \nabla G_P \) is a direction vector for the tangent line to \( C \) at \( P \).
(b) Let \( C \) be the curve of intersection of the spheres \( x^2 + y^2 + z^2 = 3 \) and \( (x-2)^2 + (y-2)^2 + z^2 = 3 \). Use the result of part (a) to find parametric equations of the tangent line to \( C \) at \( P = (1,1,1) \).
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