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Magazine Chapter 13: Problem 7
\(7-14\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=\langle\sin t, t\rangle $$
Short Answer
Expert verified
The curve follows an upward-oscillating sine wave, with arrows indicating increasing \( t \).
Step by step solution
01
Understand the Vector Equation
The vector equation given is \( \mathbf{r}(t) = \langle \sin t, t \rangle \). This represents a parametric curve where the x-coordinate is \( \sin t \), and the y-coordinate is \( t \). As \( t \) varies, these coordinates describe a curve in the xy-plane.
02
Analyze the Behavior of the Curve Components
The x-component, \( x(t) = \sin t \), oscillates between -1 and 1. The y-component, \( y(t) = t \), is linearly increasing. Thus, as \( t \) increases, the curve will oscillate in the x-direction while moving steadily upward in the y-direction.
03
Plot Key Points on the Curve
Calculate a few key points: when \( t = 0 \), \( \mathbf{r}(0) = \langle 0, 0 \rangle \); when \( t = \frac{\pi}{2} \), \( \mathbf{r}\left(\frac{\pi}{2}\right) = \langle 1, \frac{\pi}{2} \rangle \); when \( t = \pi \), \( \mathbf{r}(\pi) = \langle 0, \pi \rangle \); and when \( t = \frac{3\pi}{2} \), \( \mathbf{r}\left(\frac{3\pi}{2}\right) = \langle -1, \frac{3\pi}{2} \rangle \). Plot these points to get a sense of the curve's oscillating path.
04
Sketch the Curve
Begin sketching the curve, starting at the origin (0,0). Since the x-component oscillates between -1 and 1, the curve will appear as a sine wave that goes upward as \( t \) increases, from one key point to the next along the y-axis.
05
Indicate the Direction of Increasing \(t\)
Add an arrow to the curve to show the direction of increasing \( t \). This direction will move from left to right at each oscillation, and upward as \( t \) increases.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vector Equation
A vector equation is an expression that represents a curve or line in terms of a parameter, typically denoted as \( t \). In this context, the vector equation given is \( \mathbf{r}(t) = \langle \sin t, t \rangle \). This means:
- The curve is defined in a plane, where its path is traced as the parameter \( t \) changes.
- The x-coordinate of the curve is described by the function \( \sin t \), which tells us how the curve behaves horizontally.
- The y-coordinate is simply \( t \) itself, describing the vertical component linearly.
Oscillation
The term 'oscillation' in the context of the given vector equation \( \mathbf{r}(t) = \langle \sin t, t \rangle \) refers to the repeating motion back and forth or up and down. When we say the x-component \( \sin t \) oscillates, it means:
- The x-values constantly swing between -1 and 1. This is due to the nature of the sine function, which is periodic.
- This oscillatory behavior leads the curve to move in a wave-like manner along the x-axis as \( t \) increases.
Coordinate System
The coordinate system serves as the backdrop against which the curve defined by the vector equation \( \mathbf{r}(t) = \langle \sin t, t \rangle \) is plotted. Here's how it works:
- We use an xy-plane with the x-axis representing the horizontal component \( \sin t \), and the y-axis representing the vertical component \( t \).
- The points on the curve are determined by pairing each x-value from \( \sin t \) with a corresponding y-value from \( t \), creating coordinates \( (\sin t, t) \).
- This system allows us to visualize the relationship and motion of the curve in a two-dimensional space. It effectively shows the path traced by the changing parameter \( t \).
Curve Sketching
Curve sketching in vector equations involves plotting points and drawing a path that matches the behavior described by the equation \( \mathbf{r}(t) = \langle \sin t, t \rangle \). Here鈥檚 how you can approach it:
- First, calculate and plot key points. For example, when \( t = 0 \), \( t = \frac{\pi}{2} \), and so on, determine the corresponding coordinates \( (\sin t, t) \).
- Connect these points in the wave pattern dictated by \( \sin t \), showing the curve's oscillation horizontally.
- Note the steady increase in the y-direction due to \( t \), resulting in an upward scrolling wave.
- Finally, to indicate the direction in which \( t \) increases, draw arrows along the curve. Typically, these show a left to right movement for each wave crest and continue upwards through crests.
