18c ### Problem Statement**18. Given** $\vec{r}(t) = \langle \sin t, \, t, \, \cos t \rangle$, **Find:**### ExplanationThis vector function $\vec{r}(t)$ represents a curve in three-dimensional space. The components of the vector are functions of the parameter $t$:- The x-component is $\sin t$.- The y-component is $t$.- The z-component is $\cos t$.The problem requires you to find some property or characteristic of this curve. This could include calculating derivatives, finding specific points, or determining tangent vectors or other related quantities. The text in the image reads: "c. $ \mathcal{K}(0) $"

Name: 18c ### Problem Statement**18. Given** $\vec{r}(t) = \langle \sin t,
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Description: 18c ### Problem Statement**18. Given** \(\vec{r}(t) = \langle \sin t, \, t, \, \cos t \rangle$, **Find:**### ExplanationThis vector function $\vec{r}(t)$ represents a curve in three-dimensional space. The components of the vector are functions of

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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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18c

### Problem Statement

**18. Given** $\vec{r}(t) = \langle \sin t, \, t, \, \cos t \rangle$, **Find:**

### Explanation

This vector function $\vec{r}(t)$ represents a curve in three-dimensional space. The components of the vector are functions of the parameter $t$:

- The x-component is $\sin t$.
- The y-component is $t$.
- The z-component is $\cos t$.

The problem requires you to find some property or characteristic of this curve. This could include calculating derivatives, finding specific points, or determining tangent vectors or other related quantities.

The text in the image reads: "c. $ \mathcal{K}(0) $"

Expert Solution

Step 1

It is given that $\vec{r} (t) = <\sin t, t, \cos t>$ .

Recall the formula for the curvature at t: $κ (t) = \frac{||r' (t) \times r'' (t)||}{{(||r' (t)||)}^{3}}$ .

Obtain the first derivative of $\vec{r} (t)$ as follows.

$\vec{r}' (t) = <\cos t, 1, - \sin t>$

Similarly,

$\vec{r}'' (t) = <- \sin t, 0, - \cos t>$

Compute $r' (t) \times r'' (t)$ as follows.

$r' (t) \times r'' (t) = |\begin{matrix} i & j & k \\ \cos t & 1 & - \sin t \\ - \sin t & 0 & - \cos t \end{matrix}| = i (- \cos t) - j (\cos^{2} t + \sin^{2} t) + k (\sin t) = (- \cos t) i - j + (\sin t) k = <- \cos t, 1, \sin t>$

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