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Chapter 10: Problem 9

\(5-12=\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=\langle 1, \cos t, 2 \sin t\rangle $$

Short Answer

Expert verified
The curve is a vertical ellipse at x = 1, moving counterclockwise as \( t \) increases.

Step by step solution

01

Identify the components of the vector

The given vector equation is \( \mathbf{r}(t) = \langle 1, \cos t, 2 \sin t \rangle \). This means that the curve is defined in a three-dimensional space where the x-component is a constant \( x = 1 \), the y-component is \( y = \cos t \), and the z-component is \( z = 2 \sin t \).
02

Describe the shape of the curve

Since the x-component is constant (x = 1), this suggests that the curve lies on a plane parallel to the yz-plane. Observing the other components, we note that \( y = \cos t \) and \( z = 2 \sin t \) describe an ellipse in the yz-plane, as both are periodic functions of \( t \). The ellipse has a semi-minor axis along the y-axis with length 1, and a semi-major axis along the z-axis with length 2.
03

Sketch the curve in 3D space

To sketch the curve, plot the ellipse at the plane \( x = 1 \), which takes its form due to the trigonometric nature of \( \cos t \) and \( 2 \sin t \). The curve will look like a vertical ellipse at x = 1, parallel to the yz-plane, extending from y = -1 to y = 1, and from z = -2 to z = 2.
04

Determine the direction of the curve

The direction in which \( t \) increases is determined by the behavior of trigonometric functions. As \( t \) increases from 0 to \( 2\pi \), \( \cos t \) starts from 1, decreases to -1, and returns to 1, while \( 2\sin t \) starts from 0, goes to 2, decreases to -2, and returns to 0. Therefore, \( t \) increases in a counterclockwise direction when looking in the yz-plane.
05

Indicate the direction with an arrow

Finally, on the sketch, draw an arrow along the ellipse to denote the increasing direction of \( t \). This arrow should point in a counterclockwise direction when viewed from the positive side of the x-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

3D space sketching
When sketching in a three-dimensional space, it's essential to understand how the vector components interact. The given vector equation \( \mathbf{r}(t) = \langle 1, \cos t, 2 \sin t \rangle \) shows that the curve is set in 3D. Here, the x-component is always \( 1 \), suggesting the curve does not stretch or compress in the x-direction. Instead, it is confined to a plane where \( x = 1 \). The components \( \cos t \) and \( 2 \sin t \) determine the variation within the yz-plane.
  • The constant x-component leads to a slice of the 3D space that runs parallel to the yz-plane.
  • In this slice, the curve's shape is dictated by the functions \( \cos t \) and \( 2 \sin t \).
  • Visualizing this involves imagining a vertical plane at \( x = 1 \) through which you place the curve dictated by the yz-components.
Therefore, sketching in 3D space necessitates translating these mathematical expressions into visual forms, identifying constraints, and plotting within those boundaries.
trigonometric functions
Trigonometric functions like \( \cos t \) and \( \sin t \) are periodic functions, meaning they repeat their values in a cyclical pattern as \( t \) varies. In our vector equation, these functions determine the y and z components, creating unique movements as \( t \) increases.
  • The function \( \cos t \) varies smoothly between 1 and -1. It reaches its maximum at \( t = 0 \), minimum at \( t = \pi \), and returns to maximum at \( t = 2\pi \).
  • Meanwhile, \( 2 \sin t \) varies smoothly between 2 and -2. It starts at 0, peaks at 2 when \( t = \frac{\pi}{2} \), drops to -2 at \( t = \frac{3\pi}{2} \), and returns to 0 at \( t = 2\pi \).
These characteristics form the foundation of the ellipse shape in the yz-plane, with \( \cos t \) and \( 2 \sin t \) alternating to create symmetric stretches and compressions mirrored around the center of the ellipse.
ellipse in yz-plane
An ellipse in the yz-plane manifests from the periodic changes driven by the trigonometric components \( y = \cos t \) and \( z = 2 \sin t \). The curve demonstrates distinct characteristics of an ellipse, with axes determined by these trigonometric functions:
  • The y-axis, affected by \( \cos t \), forms a semi-minor axis with a length of 1, coinciding with the amplitude range from -1 to 1.
  • Similarly, the z-axis impacted by \( 2 \sin t \) creates a semi-major axis extending from -2 to 2, doubling the extend compared to the y-axis due to the multiplier of 2 applied to \( \sin t \).
  • This configuration indicates an ellipse centered at \( (0, 0) \) in the yz-plane.
In plotting such an ellipse, understanding the direction of \( t \) is vital. As \( t \) incrementally increases, both \( \cos t \) and \( 2 \sin t \) follow counterclockwise cycles, thus shaping and orienting the ellipse within the vertical plane at \( x = 1 \).

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