Find the unit tangent vector T and the principal unit normal vector N for the following parameterized curve. Verify that |T|=|N| = 1 and T.N=0. r(t)= (8 cos ²,8 sin t²) for 0 ≤t≤2 T=CD N=
Find the unit tangent vector T and the principal unit normal vector N for the following parameterized curve. Verify that |T|=|N| = 1 and T.N=0. r(t)= (8 cos ²,8 sin t²) for 0 ≤t≤2 T=CD N=
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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![**Problem Statement:**
Find the unit tangent vector \( \mathbf{T} \) and the principal unit normal vector \( \mathbf{N} \) for the following parameterized curve. Verify that \( |\mathbf{T}| = |\mathbf{N}| = 1 \) and \( \mathbf{T} \cdot \mathbf{N} = 0 \).
Given curve:
\[ \mathbf{r}(t) = \langle 8 \cos t^2, 8 \sin t^2 \rangle \]
for \( 0 \leq t \leq 2\pi \).
**Required:**
Compute vectors:
\[ \mathbf{T} = \langle \, \, \rangle \]
\[ \mathbf{N} = \langle \, \, \rangle \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F14ca10a7-9a5a-4024-afbf-f0f142f43830%2F8d4e6d72-ef13-4e8e-a9f1-635651a7a956%2Fwv9tgy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the unit tangent vector \( \mathbf{T} \) and the principal unit normal vector \( \mathbf{N} \) for the following parameterized curve. Verify that \( |\mathbf{T}| = |\mathbf{N}| = 1 \) and \( \mathbf{T} \cdot \mathbf{N} = 0 \).
Given curve:
\[ \mathbf{r}(t) = \langle 8 \cos t^2, 8 \sin t^2 \rangle \]
for \( 0 \leq t \leq 2\pi \).
**Required:**
Compute vectors:
\[ \mathbf{T} = \langle \, \, \rangle \]
\[ \mathbf{N} = \langle \, \, \rangle \]
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