Find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. r(t) = (3t cos t)i + (3t sin t)j + (2√/2)₁³/²k Ostst
Find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. r(t) = (3t cos t)i + (3t sin t)j + (2√/2)₁³/²k Ostst
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
![**Problem Statement:**
Find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve.
**Vector Function:**
\[ \mathbf{r}(t) = (3t \cos t) \, \mathbf{i} + (3t \sin t) \, \mathbf{j} + \left( 2 \sqrt{2} \right) t^{3/2} \, \mathbf{k} \]
**Interval:**
\[ 0 \leq t \leq \pi \]
**Explanation:**
This problem involves determining two key aspects of a parametric curve in three-dimensional space:
1. **Unit Tangent Vector:** Calculate the unit tangent vector of the given curve, \(\mathbf{r}(t)\). This requires differentiation of \(\mathbf{r}(t)\) and normalization of the resulting tangent vector.
2. **Length of the Indicated Portion:** Calculate the arc length of the curve from \(t = 0\) to \(t = \pi\). This involves integrating the magnitude of the derivative of \(\mathbf{r}(t)\) over the given interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F671be518-f9e6-4df9-9304-33791fefeccb%2F7db2256e-bb46-4d49-a85e-1219c88207e3%2Fzfh07u_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve.
**Vector Function:**
\[ \mathbf{r}(t) = (3t \cos t) \, \mathbf{i} + (3t \sin t) \, \mathbf{j} + \left( 2 \sqrt{2} \right) t^{3/2} \, \mathbf{k} \]
**Interval:**
\[ 0 \leq t \leq \pi \]
**Explanation:**
This problem involves determining two key aspects of a parametric curve in three-dimensional space:
1. **Unit Tangent Vector:** Calculate the unit tangent vector of the given curve, \(\mathbf{r}(t)\). This requires differentiation of \(\mathbf{r}(t)\) and normalization of the resulting tangent vector.
2. **Length of the Indicated Portion:** Calculate the arc length of the curve from \(t = 0\) to \(t = \pi\). This involves integrating the magnitude of the derivative of \(\mathbf{r}(t)\) over the given interval.
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