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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question

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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