Problem 7 - Advanced Engineering Mathematics

Question

In Problems, graph the curve traced by the given vector function. \(\mathbf{r}(t)=\langle\sqrt{2} \sin t, \sqrt{2} \sin t, 2 \cos t\rangle ; 0 \leq t \leq \pi / 2\)

Step-by-Step Solution

Answer

The curve is part of a cylinder along \( x = y \) with a circular cross-section when \( z \) is varied from 2 to 0.

Step 1: Understanding the Vector Function

The given vector function is \( \mathbf{r}(t) = \langle \sqrt{2} \sin t, \sqrt{2} \sin t, 2 \cos t \rangle \). This vector function describes a curve in 3D space for values of \( t \) ranging from 0 to \( \pi/2 \). Each component represents the \( x \), \( y \), and \( z \) coordinates, respectively.

Step 2: Simplifying Expressions

Notice that the \( x \) and \( y \) components are equal: \( x = \sqrt{2} \sin t \) and \( y = \sqrt{2} \sin t \). Thus, the points lie on the line \( x = y \).

Step 3: Relating to a Circle

For the \( z \) component, the expression is \( z = 2 \cos t \). Consider the identity \( \sin^2 t + \cos^2 t = 1 \). Substitute: \( x^2 + y^2 = 2 \sin^2 t = 2(1 - \cos^2 t) \).

Step 4: Deriving the Circle Equation

Substituting \( z = 2 \cos t \) into the equation derived from \( \sin^2 t + \cos^2 t = 1 \), we get \( x^2 + y^2 = 2 - z^2/2 \). Putting this in an equation of a circle, we see that \( x^2 + y^2 + \frac{z^2}{2} = 2 \).

Step 5: Drawing and Interpreting the Curve

Within the interval \( 0 \leq t \leq \pi/2 \), \( \sin t \) and \( \cos t \) transition from 0 to 1 and 1 to 0, respectively. The combination of points forms a quarter of a circle traced in the \( yz \)-plane and revolved around the line \( x = y \).

Step 6: Graphing the Curve

Graph the curve by plotting the parametrized values. Start with \( t = 0 \), which gives the point \( (0, 0, 2) \), and end with \( t = \frac{\pi}{2} \) giving \( (\sqrt{2}, \sqrt{2}, 0) \). The curve traces along the surface in the fourth quadrant of the \( yz \)-plane excluding axes.

Key Concepts

3D Space

3D space is an environment where any point can be defined by three coordinates: typically denoted as \(x\), \(y\), and \(z\). This three-dimensional coordinate system, often called the Cartesian coordinate system, allows us to locate any point in space through a combination of these coordinates.

Each coordinate represents a specific dimension:

• \(x\) often refers to the horizontal axis,
• \(y\) represents the vertical axis, and
• \(z\) is the depth axis.

In this kind of space, we can visualize complex geometries and paths by using mathematical representations, like vector functions. These provide a systematic way of following trajectories and locating points accurately.

In our scenario, the given vector function describes a curve traced in this 3D space. Each component of the vector function provides us with the respective \(x\), \(y\), and \(z\) positions as the parameter \(t\) changes, giving the path a tangible form in 3D geometry.

Parametric Equations

Parametric equations are an essential mathematical tool for describing curves and shapes, especially in higher dimensions. Instead of defining \(y\) directly as a function of \(x\), parametric equations use an independent parameter—often denoted as \(t\)—to express both \(x\) and \(y\). In 3D space, they extend to include the \(z\) variable as well.

The given vector function \( \mathbf{r}(t) = \langle \sqrt{2} \sin t, \sqrt{2} \sin t, 2 \cos t \rangle \) is parametric. It uses \(t\):

• \(x(t) = \sqrt{2} \sin t\)
• \(y(t) = \sqrt{2} \sin t\)
• \(z(t) = 2 \cos t\)

This setup lets us explore how the curve behaves as \(t\) varies within the range \(0 \leq t \leq \pi/2\). It's easy to track how each coordinate changes with \(t\) and thus understand the curve's geometry or path.Through parametric equations, we comprehend interactions within 3D space more naturally, offering insights into not only the curve's shape but also its orientation and position.

Graphing Curves

To graph curves described by vector functions like \( \mathbf{r}(t) = \langle \sqrt{2} \sin t, \sqrt{2} \sin t, 2 \cos t \rangle \), you need to visualize how the parametric equations work together. These curves can represent paths or trajectories followed in 3D space, essential for understanding more complex motion and shapes.

Here's how to approach graphing such a curve:

• Begin with setting \(t = 0\). Substitute into the equations to find the starting point. For this function, it's \((0, 0, 2)\).
• As \(t\) increases to \(\frac{\pi}{2}\), substitute to find other key points. Here, it reaches \((\sqrt{2}, \sqrt{2}, 0)\).
• Recognize patterns and relationships, such as the ones depicting that \(x = y\), which indicates part of the curve lies along this line.
• Look for any familiar shapes in the planes, like portions of circles or ellipses, when observing the undulating | and revolving pattern formed by \(z\).

This method provides a concrete graphical representation of the curve within the defined boundaries of \(t\). Graphing gives not only visual insight but also reinforces understanding of the curve’s nature and the underlying theory of vector functions.