Let r(t)=r(t)=e¹i+e=¹j+tk. (a) Find the unit tangent, normal and binormal vectors T(t), N(t) and B(t) at t=0. (h) Find
Let r(t)=r(t)=e¹i+e=¹j+tk. (a) Find the unit tangent, normal and binormal vectors T(t), N(t) and B(t) at t=0. (h) Find
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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![### Curvature and Torsion of a Vector Function
Let \(\mathbf{r}(t) = e^{t} \mathbf{i} + e^{-t} \mathbf{j} + t \mathbf{k}\).
#### Task:
(a) Find the unit tangent, normal, and binormal vectors \(\mathbf{T}(t)\), \(\mathbf{N}(t)\), and \(\mathbf{B}(t)\) at \(t = 0\).
(b) Find the curvature at \(t = 0\).
**Explanation and Solution Steps:**
1. **Unit Tangent Vector, \(\mathbf{T}(t)\)**:
The unit tangent vector is given by:
\[
\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{\|\mathbf{r}'(t)\|}
\]
2. **Unit Normal Vector, \(\mathbf{N}(t)\)**:
The unit normal vector is given by:
\[
\mathbf{N}(t) = \frac{\mathbf{T}'(t)}{\|\mathbf{T}'(t)\|}
\]
3. **Binormal Vector, \(\mathbf{B}(t)\)**:
The binormal vector is given by:
\[
\mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t)
\]
4. **Curvature, \(\kappa(t)\)**:
The curvature is given by:
\[
\kappa(t) = \frac{\|\mathbf{r}'(t) \times \mathbf{r}''(t)\|}{\|\mathbf{r}'(t)\|^3}
\]
### Solution at \(t = 0\):
- **Find \(\mathbf{r}'(t)\), \(\mathbf{r}''(t)\)**:
\[
\mathbf{r}'(t) = \frac{d}{dt} (e^{t} \mathbf{i} + e^{-t} \mathbf{j} + t \mathbf{k}) = e^{t} \mathbf{i} - e^{-t} \mathbf{j} + \mathbf{k}
\]
\[
\mathbf{r}''(t) = \frac{d}{dt} \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F304238c1-892d-4f66-90f4-8d5405bfbe58%2Fff6737ed-4524-4bc8-8a9b-2600196b1e90%2Fhpguzdf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Curvature and Torsion of a Vector Function
Let \(\mathbf{r}(t) = e^{t} \mathbf{i} + e^{-t} \mathbf{j} + t \mathbf{k}\).
#### Task:
(a) Find the unit tangent, normal, and binormal vectors \(\mathbf{T}(t)\), \(\mathbf{N}(t)\), and \(\mathbf{B}(t)\) at \(t = 0\).
(b) Find the curvature at \(t = 0\).
**Explanation and Solution Steps:**
1. **Unit Tangent Vector, \(\mathbf{T}(t)\)**:
The unit tangent vector is given by:
\[
\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{\|\mathbf{r}'(t)\|}
\]
2. **Unit Normal Vector, \(\mathbf{N}(t)\)**:
The unit normal vector is given by:
\[
\mathbf{N}(t) = \frac{\mathbf{T}'(t)}{\|\mathbf{T}'(t)\|}
\]
3. **Binormal Vector, \(\mathbf{B}(t)\)**:
The binormal vector is given by:
\[
\mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t)
\]
4. **Curvature, \(\kappa(t)\)**:
The curvature is given by:
\[
\kappa(t) = \frac{\|\mathbf{r}'(t) \times \mathbf{r}''(t)\|}{\|\mathbf{r}'(t)\|^3}
\]
### Solution at \(t = 0\):
- **Find \(\mathbf{r}'(t)\), \(\mathbf{r}''(t)\)**:
\[
\mathbf{r}'(t) = \frac{d}{dt} (e^{t} \mathbf{i} + e^{-t} \mathbf{j} + t \mathbf{k}) = e^{t} \mathbf{i} - e^{-t} \mathbf{j} + \mathbf{k}
\]
\[
\mathbf{r}''(t) = \frac{d}{dt} \
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