r(t) = (sin 7t, t, cos t)
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
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases.
![**Problem 10:**
The vector function \( \mathbf{r}(t) \) is given by:
\[
\mathbf{r}(t) = \langle \sin(\pi t), t, \cos(\pi t) \rangle
\]
**Explanation:**
This represents a parametric curve in three-dimensional space where:
- The \(x\)-component is \(\sin(\pi t)\).
- The \(y\)-component is \(t\).
- The \(z\)-component is \(\cos(\pi t)\).
As \(t\) varies, the point \((\sin(\pi t), t, \cos(\pi t))\) traces a path through space. The curve intertwines sinusoidal motion in the \(x\)- and \(z\)-directions with a linear increase in the \(y\)-direction.
This curve can be visualized as an ascending trajectory with sinusoidal oscillations in two axes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F160e110e-0605-48f5-87d6-7df540ee4abb%2F06beef43-97c3-454a-ae68-13126358769e%2Flhpbav9.png&w=3840&q=75)
Transcribed Image Text:**Problem 10:**
The vector function \( \mathbf{r}(t) \) is given by:
\[
\mathbf{r}(t) = \langle \sin(\pi t), t, \cos(\pi t) \rangle
\]
**Explanation:**
This represents a parametric curve in three-dimensional space where:
- The \(x\)-component is \(\sin(\pi t)\).
- The \(y\)-component is \(t\).
- The \(z\)-component is \(\cos(\pi t)\).
As \(t\) varies, the point \((\sin(\pi t), t, \cos(\pi t))\) traces a path through space. The curve intertwines sinusoidal motion in the \(x\)- and \(z\)-directions with a linear increase in the \(y\)-direction.
This curve can be visualized as an ascending trajectory with sinusoidal oscillations in two axes.
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