Problem 9 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

The curve forms a 3D spiral moving upward.

Step 1: 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 \( t \).

Step 2: 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.

Step 3: 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.

Key Concepts

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.