Problem 4 - Advanced Engineering Mathematics

Question

In Problems, graph the curve traced by the given vector function. \(\mathbf{r}(t)=4 \mathbf{i}+2 \cos t \mathbf{j}+3 \sin t \mathbf{k}\)

Step-by-Step Solution

Answer

The curve is an ellipse in the plane \( x = 4 \), with axes 2 and 3 along the \( y \) and \( z \) directions.

Step 1: Understand the Vector Function

The given vector function is \( \mathbf{r}(t) = 4 \mathbf{i} + 2 \cos t \mathbf{j} + 3 \sin t \mathbf{k} \). This means for any value of \( t \), the vector function provides a point \( (x(t), y(t), z(t)) \) in 3D space, where \( x(t) = 4 \), \( y(t) = 2 \cos t \), and \( z(t) = 3 \sin t \).

Step 2: Analyze the Components

The \( x(t) \) component is constant at 4, meaning the curve is aligned along the plane \( x = 4 \). The \( y(t) \) and \( z(t) \) components, \( 2 \cos t \) and \( 3 \sin t \), describe an ellipse in the \( yz \)-plane because cosine and sine functions form circular motion, but are scaled by different coefficients (2 and 3).

Step 3: Parametrize the Curve

To graph the curve, see how the ellipse formed by \( y(t) = 2 \cos t \) and \( z(t) = 3 \sin t \) behaves. Since \( t \) is a parameter, as \( t \) varies from 0 to \( 2\pi \), the ellipse will be traced in the \( yz \)-plane. The center of this ellipse is at \( (4, 0, 0) \), and it has a semi-major axis of 3 along the \( z \)-axis and a semi-minor axis of 2 along the \( y \)-axis.

Step 4: Visualize the Graph

Since the curve is on the plane \( x = 4 \), you graph an ellipse centered at \( (4,0,0) \) with major and minor axes determined by \( 3 \sin t \) and \( 2 \cos t \) respectively. The ellipse is contained entirely within the \( yz \)-plane parallel to the \( x \)-axis.

Key Concepts

Vector Functions

Vector functions are a blend of vectors and functions, and they map real numbers to multi-dimensional quantities. In this case, we consider \( \mathbf{r}(t) = 4 \mathbf{i} + 2 \cos t \mathbf{j} + 3 \sin t \mathbf{k} \).This means for every scalar input \(t\), the function will give us a vector with components in the three dimensions of space.

• \(x(t) = 4\), which is constant.
• \(y(t) = 2 \cos t\)
• \(z(t) = 3 \sin t\)

Understanding these components allows us to know where the point will be in 3D space for any given \(t\).
The vector function essentially traces a path in this 3D space as \(t\) varies.

3D Space

3D space is a mathematical setting where each point is specified by three coordinates (\(x, y, z\)). In this problem, each point is part of the function \( \mathbf{r}(t) \) which assigns a specific position in 3D space depending on \(t\).

• The coordinate system uses three axes: \(x\), \(y\), and \(z\).
• Every point on the graph corresponds to a particular combination of these coordinates.
• The curve is essentially a path traced by the continuous points defined by the vector function.

The beauty of 3D space lies in its ability to represent real-world scenarios more effectively than 2D representations.
Moving in 3D allows objects to have planes and volumes, adding depth to their visualization.

Ellipse Parametric Equations

Ellipses in mathematics can be represented using parametric equations. For this exercise, the parametric equations describe the curve traced out in the \( yz\)-plane while keeping \(x=4\) constant.
The equations are:

• \(y(t) = 2 \cos t\)
• \(z(t) = 3 \sin t\)

These parametric equations provide a straightforward way to define an ellipse.
The parameters are coefficients of the sine and cosine functions, which determine the size of the ellipse.
In the parametric form, each oscillates back and forth, mimicking natural elliptical motion.
Ellipses are oblong circles whose lengths and widths are determined by these coefficients, representing semi-major and semi-minor axes.

Graphing Techniques

Graphing in 3D can feel overwhelming, but breaking it down step by step eases the process. Let's see how you would proceed in this scenario:
1. **Identify the Planes**: Recognize which plane the movement takes place in. Here, it's the \( yz\)-plane.2. **Determine Fixed Values**: Notice the constant \(x=4\). This fixates the entire path at this \(x\) value.3. **Plot Ellipse**: Using the parametric equations for \(y\) and \(z\), plot points that represent the ellipse.

• The semi-major axis is 3, mapped to \(z\).
• The semi-minor axis is 2, associated with \(y\).

4. **Use Technology**: For more precision, graphing tools or software can help visualize the curve accurately.
Realizing that practice is key to mastering these graphing techniques enhances understanding and builds confidence when working with vector functions in 3D space.