Find the unit tangent vector T and the curvature k for the following parameterized curve. r(t) = (V51 cos t,7 cos t 10 sin t) TE... The unit tangent vector is T= (Type exact answers, using radicals as needed.)
Find the unit tangent vector T and the curvature k for the following parameterized curve. r(t) = (V51 cos t,7 cos t 10 sin t) TE... The unit tangent vector is T= (Type exact answers, using radicals as needed.)
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
Unit tangent
the curvature is k=?
![**Finding the Unit Tangent Vector and Curvature**
**Problem Statement:**
Find the unit tangent vector \( \mathbf{T} \) and the curvature \( \kappa \) for the following parameterized curve:
\[
\mathbf{r}(t) = \langle \sqrt{51} \cos t, 7 \cos t, 10 \sin t \rangle
\]
**Solution Process:**
1. **Unit Tangent Vector \( \mathbf{T} \):**
The unit tangent vector is denoted as \( \mathbf{T} \) and can be found by taking the derivative of \( \mathbf{r}(t) \) with respect to \( t \), then normalizing it.
\[
\mathbf{T} = \langle \, \text{Type exact answers, using radicals as needed.} \, \rangle
\]
2. **Steps to Solve:**
- **Calculate the derivative** \( \mathbf{r}'(t) = \frac{d}{dt}\mathbf{r}(t) \).
- **Find the magnitude** of \( \mathbf{r}'(t) \).
- **Normalize** \( \mathbf{r}'(t) \) by dividing by its magnitude to get \( \mathbf{T} \).
**Note:**
This exercise requires you to express your answers exactly, using radicals where necessary.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca46498c-9039-4635-914d-0b10c3becf43%2F65748e6b-6b17-429f-b57a-568a025a79ab%2Fqe2xmh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Unit Tangent Vector and Curvature**
**Problem Statement:**
Find the unit tangent vector \( \mathbf{T} \) and the curvature \( \kappa \) for the following parameterized curve:
\[
\mathbf{r}(t) = \langle \sqrt{51} \cos t, 7 \cos t, 10 \sin t \rangle
\]
**Solution Process:**
1. **Unit Tangent Vector \( \mathbf{T} \):**
The unit tangent vector is denoted as \( \mathbf{T} \) and can be found by taking the derivative of \( \mathbf{r}(t) \) with respect to \( t \), then normalizing it.
\[
\mathbf{T} = \langle \, \text{Type exact answers, using radicals as needed.} \, \rangle
\]
2. **Steps to Solve:**
- **Calculate the derivative** \( \mathbf{r}'(t) = \frac{d}{dt}\mathbf{r}(t) \).
- **Find the magnitude** of \( \mathbf{r}'(t) \).
- **Normalize** \( \mathbf{r}'(t) \) by dividing by its magnitude to get \( \mathbf{T} \).
**Note:**
This exercise requires you to express your answers exactly, using radicals where necessary.
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