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(b) Let C₁, C₂ and C3 be the curves in R³ with parametrisations C₁: (SER) (s − 2)i + (2s² − 1)j + k 2t² i + (3t+7)j + (t+1) k C₂ (t ≤ R) C3 (u² + 4u + 4) i + (u² + 6)j + (u + 2) k_(u≤R) respectively (i) r₁(s) r₂(t) r3(u) = = = giving that the point of intersection exist and just asking you to verify that the curves intersect at a point answer. (ii) Find the tangent vectors to the curves C₁, C2 and C3 at the unique point of intersection you find in part (i). Do these tangent vectors lie in a common plane in R³? Justify your (iii) Is it possible to find a surface z = h(x, y) for which the curves C₁, C₂ and C3 all lie on this surface? Justify your answer.

(b) Let C₁, C₂ and C3 be the curves in R³ with parametrisations C₁: (SER) (s − 2)i + (2s² − 1)j + k 2t² i + (3t+7)j + (t+1) k C₂ (t ≤ R) C3 (u² + 4u + 4) i + (u² + 6)j + (u + 2) k_(u≤R) respectively (i) r₁(s) r₂(t) r3(u) = = = giving that the point of intersection exist and just asking you to verify that the curves intersect at a point answer. (ii) Find the tangent vectors to the curves C₁, C2 and C3 at the unique point of intersection you find in part (i). Do these tangent vectors lie in a common plane in R³? Justify your (iii) Is it possible to find a surface z = h(x, y) for which the curves C₁, C₂ and C3 all lie on this surface? Justify your answer.

Advanced Engineering Mathematics
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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(b) Let C₁, C₂ and C3 be the curves in R³ with parametrisations C₁: r₁(s) (s − 2)i + (2s² − 1)j + k - C₂ r₂(t) C3 r3(u) respectively (i) = = (SER) 2t² i + (3t+7)j + (t+1) k (t = R) (u² +4u + 4) i + (u² + 6)j + (u + 2) k (u ¤R) giving that the point of intersection exist and just asking you to verify that the curves intersect at a point (ii) Find the tangent vectors to the curves C₁, C2 and C3 at the unique point of intersection you find in part (i). Do these tangent vectors lie in a common plane in R³? Justify your answer. (iii) Is it possible to find a surface z = h(x, y) for which the curves C₁, C₂ and C3 all lie on this surface? Justify your answer.
Transcribed Image Text:(b) Let C₁, C₂ and C3 be the curves in R³ with parametrisations C₁: r₁(s) (s − 2)i + (2s² − 1)j + k - C₂ r₂(t) C3 r3(u) respectively (i) = = (SER) 2t² i + (3t+7)j + (t+1) k (t = R) (u² +4u + 4) i + (u² + 6)j + (u + 2) k (u ¤R) giving that the point of intersection exist and just asking you to verify that the curves intersect at a point (ii) Find the tangent vectors to the curves C₁, C2 and C3 at the unique point of intersection you find in part (i). Do these tangent vectors lie in a common plane in R³? Justify your answer. (iii) Is it possible to find a surface z = h(x, y) for which the curves C₁, C₂ and C3 all lie on this surface? Justify your answer.
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