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se the given curve to do the following calculations. r(t)=2ti+ 4 costj+4 sint k, for 0 st≤2 . Find the tangent vector and the unit tangent vector. D. Find the curvature. C. Find the principal unit normal vector

se the given curve to do the following calculations. r(t)=2ti+ 4 costj+4 sint k, for 0 st≤2 . Find the tangent vector and the unit tangent vector. D. Find the curvature. C. Find the principal unit normal vector

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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Use the given curve to do the following calculations. r(t) = 2ti+ 4 costj+4 sint k, for 0 st≤2 a. Find the tangent vector and the unit tangent vector. b. Find the curvature. c. Find the principal unit normal vector. d. Verify that |N|= 1 and T.N=0. e. Graph the curve and sketch T and N at two points. a. The tangent vector is r'(t)= (2-4 sint. 4 cos t The unit tangent vector is T(t)=0- b. The curvature is x = (Simplify your answer.) c. The principal unit normal vector is N(t) = (0 - cos d. Verify that N=1 and T-N=0. Verify that N=1, where N(t)= (x(t),y(t).z(t)). Select the correct choice below and fill in the (Type the terms of your expression in the same order as they appear in the original expressi OA. IN = x(t)² + y(t)² + z(t)² = OB. IN (√xt) + √y(t) + √/z(t))² C. IN = √x(1)² + y(t)². + z(t) = + + COS + + - sin t + sin t Verify that T-N=0. Select the correct choice below and fill in the answer boxes to complete y (Type the terms of your expression in the same order as they appear in the original expression OA. T•N=u₂v₁ +₂V₂+U₂V₂ = (1 =0 OB. T•N=-U₁V OCT•N=u₂v₁ U₂V₂ + Uy Vy= e. Choose the correct plot below. Click here to view plot.c Click here to view plot b = 1 =0 2 =1 =0 =1
Transcribed Image Text:Use the given curve to do the following calculations. r(t) = 2ti+ 4 costj+4 sint k, for 0 st≤2 a. Find the tangent vector and the unit tangent vector. b. Find the curvature. c. Find the principal unit normal vector. d. Verify that |N|= 1 and T.N=0. e. Graph the curve and sketch T and N at two points. a. The tangent vector is r'(t)= (2-4 sint. 4 cos t The unit tangent vector is T(t)=0- b. The curvature is x = (Simplify your answer.) c. The principal unit normal vector is N(t) = (0 - cos d. Verify that N=1 and T-N=0. Verify that N=1, where N(t)= (x(t),y(t).z(t)). Select the correct choice below and fill in the (Type the terms of your expression in the same order as they appear in the original expressi OA. IN = x(t)² + y(t)² + z(t)² = OB. IN (√xt) + √y(t) + √/z(t))² C. IN = √x(1)² + y(t)². + z(t) = + + COS + + - sin t + sin t Verify that T-N=0. Select the correct choice below and fill in the answer boxes to complete y (Type the terms of your expression in the same order as they appear in the original expression OA. T•N=u₂v₁ +₂V₂+U₂V₂ = (1 =0 OB. T•N=-U₁V OCT•N=u₂v₁ U₂V₂ + Uy Vy= e. Choose the correct plot below. Click here to view plot.c Click here to view plot b = 1 =0 2 =1 =0 =1
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