Question

Consider the curve \(\mathbf{r}(t)=(a \cos t+b \sin t) \mathbf{i}+(c \cos t+d \sin t) \mathbf{j}+(e \cos t+f \sin t) \mathbf{k}\) where \(a, b, c, d, e,\) and fare real numbers. It can be shown that this curve lies in a plane. Graph the following curve and describe it. $$\begin{aligned} \mathbf{r}(t)=&(2 \cos t+2 \sin t) \mathbf{i}+(-\cos t+2 \sin t) \mathbf{j} \\\ &+(\cos t-2 \sin t) \mathbf{k} \end{aligned}$$

Short answer

Question: Graph and describe the curve given by the vector function $$\mathbf{r}(t)=(2 \cos t+2 \sin t) \mathbf{i}+(-\cos t+2 \sin t) \mathbf{j}+(\cos t-2 \sin t) \mathbf{k}$$ where t lies within the range [0, 2π].

Step-by-step solution

Parametrize and identify components of the curve

The given vector function for the curve is: $$\mathbf{r}(t)=(2 \cos t+2 \sin t) \mathbf{i}+(-\cos t+2 \sin t) \mathbf{j}+(\cos t-2 \sin t) \mathbf{k}$$ From the expression, we can identify the x, y, and z components of the curve as: $$x(t)=2 \cos t+2 \sin t$$ $$y(t)=-\cos t+2 \sin t$$ $$z(t)=\cos t-2 \sin t$$

Choose a range for the parameter t

Since the curve is periodic due to the presence of sin and cos functions, we can set the range of t as [0, 2π]. This will enable us to have a complete visualization of the curve. Let us choose multiple points within this range and evaluate x(t), y(t), and z(t) to generate points on the curve.

Evaluate x, y, and z component at chosen t values

Evaluate x(t), y(t), and z(t) for some chosen values of t between 0 and 2π. For example, you can choose t values: 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4, and 2π. For each chosen value of t, plug it into the formula for x(t), y(t), and z(t) to find the corresponding coordinates.

Plot the curve with the calculated points

Once you have calculated x, y, and z values for the chosen t values, proceed to plot the 3D curve. Use the maximum and minimum values of x, y, and z that you have calculated to set the axes limits. Plot the points in 3D space and connect them to visualize the curve.

Analyze and describe the curve

Based on the plotted curve, analyze its shape and general characteristics. For instance, determine if the curve is closed, if it is planar, or if it has a helical shape. Describe the curve in words, indicating its shape and any unique features that you observe. After following these steps, you will have successfully graphed and described the given curve.

Key concepts

3D Graphing

When we talk about 3D graphing, we're exploring the way we can visualize equations that involve three dimensions: x, y, and z. In the exercise, we have a curve defined by a vector function, which gives values for x, y, and z in terms of the parameter \(t\). To truly understand this curve, we need to plot it in a 3D space.

• Understanding 3D Space: Visualizing functions in three dimensions means considering how each variable interacts spatially. Imagine graphs going beyond flat paper into a structure you can twist and view from different angles.
• Plotting Points: Identify points by evaluating the vector function for different values of \(t\). Here, we've chosen \(t\) ranging from 0 to \(2\pi\) since trigonometric functions often complete a cycle within this interval. This helps visualize the entire spread of the curve.
• Connecting the Dots: Once the points are plotted in 3D space, connect them smoothly to replicate the curve. This is how you bring the function's path to life.

Vector Functions

Vector functions are central to plotting curves like the one in our exercise. They create a connection between a scalar parameter, say \(t\), and vector values that define several coordinates:\(x(t), y(t), z(t)\). In essence, vector functions provide a way to draw paths through space.

• Components of Vector Functions: Each component - \(x(t)\), \(y(t)\), and \(z(t)\), is a function itself. Here these are defined with cosine and sine functions, showing the relationship between \(t\) and each axis.
• Planar Curves: A curve might be planar, meaning it lies in a single plane. Despite existing in 3D space, all its points fit in one flat surface. Our curve, though existing in a 3D space, is calculated to be planar based on its parametric form.
• Building Curves: By tweaking the components of a vector function, you can create different curves and paths. That’s the beauty of vector functions—they offer a mathematical toolset for 3D exploration.

Trigonometric Functions

Trigonometric functions like sine and cosine are pivotal in defining curves because they inherently involve periodicity and oscillation. In our exercise, these functions define the curve’s cyclical nature across the \(t\)-interval.

• Periodicity: Both sine and cosine functions repeat every \(2\pi\). This repeating pattern is why we choose this interval for \(t\), ensuring one complete cycle of the curve is visible.
• Sine and Cosine in Curves: By modifying coefficients, these functions control the curve's shape and orientation. Notice how varying the coefficients in front of sine and cosine affect the curve's axis alignment and oscillations.
• Analyzing Effects: In the given curve, the coefficients dictate how far and in what direction the oscillations occur along each axis. Experiment with coefficients to better grasp how these functions mold the 3D curve.