Problem 9 In Problems, graph the curve tra... [FREE SOLUTION]

Chapter 9: Problem 9

In Problems, graph the curve traced by the given vector function. \(\mathbf{r}(t)=e^{t} \cos t \mathbf{i}+e^{t} \sin t \mathbf{j}+e^{t} \mathbf{k}\)

Short Answer

Expert verified

The curve forms a 3D spiral moving upward.

Step by step solution

Understanding the Components

The vector function \( \mathbf{r}(t) = e^t \cos t \mathbf{i} + e^t \sin t \mathbf{j} + e^t \mathbf{k} \) describes a curve in three-dimensional space. It consists of three components: \( x(t) = e^t \cos t \), \( y(t) = e^t \sin t \), and \( z(t) = e^t \). These each define a position in space based on the parameter \( t \).

Analyzing the Curve

Let's examine how each component behaves: \( x(t) \) and \( y(t) \) are similar to polar coordinates \((r, \theta)\) with \( x = r \cos \theta \) and \( y = r \sin \theta \), where \( r = e^t \). This indicates a spiral because \( r \) (the distance from the origin in the xy-plane) increases as \( t \) increases. \( z(t) = e^t \) means the curve moves upwards as \( t \) increases.

Visualizing the Path

The curve is a three-dimensional spiral moving upwards. As \( t \) increases, \( e^t \) increases exponentially, causing the spiral to expand outward and rise upwards. Imagine a helicoid ascending vertically as it spirals outward.

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Key Concepts

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

Vector Function

In vector calculus, a vector function is a function where the domain is one or more variables, and the codomain is a set of vectors in space. In our example, the vector function is expressed as \( \mathbf{r}(t) = e^t \cos t \mathbf{i} + e^t \sin t \mathbf{j} + e^t \mathbf{k} \).This type of function assigns a vector to every value of the parameter \( t \).
Key points to understand about vector functions include:

Each component, \( x(t), y(t), \) and \( z(t) \), defines part of a coordinate in space.
The vector \( \mathbf{r}(t) \) traces a path or curve as \( t \) changes.
The function can describe motion or changes in a point within space.

This representation is crucial for studying objects and trajectories in physics and engineering because it provides detailed positional information.

Three-Dimensional Space

Three-dimensional space is the setting for our vector function. It's an extension of two-dimensional space, adding depth to the length and width, with coordinates typically represented as \((x, y, z)\). The vector function \( \mathbf{r}(t) \) exemplifies this concept by mapping the parameter \( t \) to a position in three-dimensional space.
Here's why understanding three-dimensional space is important:

It allows us to visualize and graphically represent objects and functions that exist in the real world.
Each coordinate axis (x, y, z) provides a means to specify positions and trajectories uniquely.
Graphs, such as our helical curve, can be constructed to represent the path of an object.

Understanding three-dimensional space is vital in many fields, such as engineering, physics, and computer graphics, where spatial awareness and trajectory prediction are key.

Helical Curves

Helical curves represent a specific type of space curve that spiral around an axis. In the example vector function, such curves are apparent. Characterized by both a circular and linear movement, they can be visualized as a thread spiraling around the core of a screw. The expression \( \mathbf{r}(t) = e^t \cos t \mathbf{i} + e^t \sin t \mathbf{j} + e^t \mathbf{k} \) outlines a helical path.
Important attributes of helical curves include:

The spiral expansion and height increase as \( t \) increases, showing an exponential growth along both horizontal and vertical axes.
The helical path continuously rotates around the vertical axis while moving upwards.
This can't be fully captured in two dimensions, exemplifying the requirement of three-dimensional space for its representation.

Helical curves are abundant in various natural phenomena and man-made structures, ranging from DNA's double helix to spiral staircases. Understanding them helps in diverse applications from molecular biology to architecture.

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91影视

In Problems, graph the curve traced by the given vector function. \(\mathbf{r}(t)=e^{t} \cos t \mathbf{i}+e^{t} \sin t \mathbf{j}+e^{t} \mathbf{k}\)

Short Answer

Step by step solution

Understanding the Components

Analyzing the Curve

Visualizing the Path

Key Concepts

Vector Function

Three-Dimensional Space

Helical Curves

One App. One Place for Learning.

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