Problem 1 - Advanced Engineering Mathematics

Question

In Problems, graph the curve traced by the given vector function. \(\mathbf{r}(t)=2 \sin t \mathbf{i}+4 \cos t \mathbf{j}+t \mathbf{k} ; t \geq 0\)

Step-by-Step Solution

Answer

The curve is a helix rising along the z-axis, with elliptical traces in the xy-plane.

Step 1: Understanding the Vector Function

First, let's analyze the given vector function \( \mathbf{r}(t) = 2 \sin t \mathbf{i} + 4 \cos t \mathbf{j} + t \mathbf{k} \). It defines a curve in 3D space, where each component function represents a coordinate: \( x(t) = 2 \sin t \), \( y(t) = 4 \cos t \), and \( z(t) = t \).

Step 2: Recognize the Parametric Equations

Recognize that the vector function can be broken down into parametric equations: \( x = 2\sin t \), \( y = 4\cos t \), and \( z = t \). This will help in determining the structure and motion of the curve.

Step 3: Identify the Base Curve Component

The equations \( x = 2\sin t \) and \( y = 4\cos t \) suggest an ellipse in the xy-plane, as they follow the format \( \frac{x^2}{4} + \frac{y^2}{16} = 1 \), by using the identity \( \sin^2 t + \cos^2 t = 1 \).

Step 4: Include the Third Dimension

The z-component \( z = t \) indicates that as \( t \) increases, the curve will move upward along the z-axis. This means the ellipse in the xy-plane is traced perpendicularly upwards as t increases.

Step 5: Graphing the Curve

To graph the curve, first plot the ellipse in the xy-plane for several values of \( t \). Next, consider that the ellipse is elevating upwards linearly with z.1. Plot points such as: When \( t = 0 \), \( \mathbf{r}(0)=(0,4,0) \); when \( t = \frac{\pi}{2} \), \( \mathbf{r}\left(\frac{\pi}{2}\right)=(2,0,\frac{\pi}{2}) \); and when \( t = \pi \), \( \mathbf{r}(\pi)=(0,-4,\pi) \).2. Connect these points considering the trajectory upwards, representing a helical shape.

Step 6: Visualize the Helix

Based on these observations, visualize the helical path, resembling a stretched spring, with each circular loop expanding into the xy-plane and moving upwards. This helix has a consistent rise per loop, correlating with the linear increase in the z-component.

Key Concepts

Parametric Equations

Parametric equations are a way to describe a curve in space using parameters, often denoted as "t". This approach involves expressing each coordinate (x, y, z) as a function of "t". In the case of our vector function \( \mathbf{r}(t) = 2 \sin t \mathbf{i} + 4 \cos t \mathbf{j} + t \mathbf{k} \), the parametric equations are:

• \( x = 2 \sin t \)
• \( y = 4 \cos t \)
• \( z = t \)

This means for every value of "t", you get a corresponding point in 3D space determined by these equations.
Parametric equations are extremely useful for modeling curves and motions that are difficult to describe using traditional Cartesian equations.
They allow us to break down complex motion into separate components, making analysis much simpler.

3D Space Curves

3D space curves are paths or trajectories that are traced by a point moving in three-dimensional space. These curves are described using vector functions such as \( \mathbf{r}(t) \).
Each component of the vector function represents movement along one of the three dimensions: x, y, or z.
In our example, the vector function describes a space curve that combines an elliptical motion in the xy-plane with a linear movement along the z-axis.
This gives rise to a more complex trajectory that cannot be captured by single-coordinate equations.
Understanding 3D space curves can help us visualize and analyze movements of objects in real-world applications, such as the trajectory of a drone flying in the sky.

Helical Motion

Helical motion is characterized by a spiral path that rises or descends in space. This type of motion is similar to the path traced by a helix or a screw.
In our function \( \mathbf{r}(t) = 2 \sin t \mathbf{i} + 4 \cos t \mathbf{j} + t \mathbf{k} \), the curve traces out a helical shape, where:

• The xy components \( 2 \sin t \) and \( 4 \cos t \) describe an ellipse.
• The z component \( z = t \) indicates linear elevation as "t" increases.

This combination creates a spiraling movement, resembling a twisted coil, with each turn rising steadily upwards.
Helical motions are common in mechanical systems such as springs and can also be observed in natural phenomena like DNA structures.

Ellipse in xy-plane

An ellipse in the xy-plane is a flattened circle with a major and minor axis. The parametric equations \( x = 2 \sin t \) and \( y = 4 \cos t \) describe such an ellipse.
This is because the equations fit the general form \( \frac{x^2}{4} + \frac{y^2}{16} = 1 \), derived from the identity \( \sin^2 t + \cos^2 t = 1 \).
The coefficients "2" and "4" define the "radii" along the x and y axes, respectively:

• "2" is the semi-minor axis along x.
• "4" is the semi-major axis along y.

By visualizing this, you get an elongated circle in the xy-plane, oriented with its longer side along the y-axis.
Ellipses are significant in various fields, including astronomy, as they describe the orbits of planets.