Problem 53 - Thomas Calculus

Question

In Exercises 53 and \(54,\) you will explore graphically the behavior of the helix $$\quad \mathbf{r}(t)=(\cos a t) \mathbf{i}+(\sin a t) \mathbf{j}+b t \mathbf{k}$$ as you change the values of the constants \(a\) and \(b .\) Use a CAS to perform the steps in each exercise. Set \(b=1 .\) Plot the helix \(\mathbf{r}(t)\) together with the tangent line to the curve at \(t=3 \pi / 2\) for \(a=1,2,4,\) and 6 over the interval \(0 \leq t \leq 4 \pi .\) Describe in your own words what happens to the graph of the helix and the position of the tangent line as \(a\) increases through these positive values.

Step-by-Step Solution

Answer

As \( a \) increases, the helix coils more tightly, and the tangent line's inclination shifts, showing more frequent revolutions.

Step 1: Understand the Helix Equation

The given helix is represented by the vector function \( \mathbf{r}(t) = (\cos at) \mathbf{i} + (\sin at) \mathbf{j} + bt \mathbf{k} \). Here, \( a \) and \( b \) are constants altering the shape of the helix. For this exercise, \( b \) is set to 1, simplifying the vertical component to \( t \).

Step 2: Compute the Tangent Line at a Specific Point

To find the tangent line at \( t = \frac{3\pi}{2} \), we need the derivative of the helix: \( \mathbf{r}'(t) = (-a \sin at) \mathbf{i} + (a \cos at) \mathbf{j} + \mathbf{k} \). Evaluate \( \mathbf{r}(t) \) and \( \mathbf{r}'(t) \) at \( t = \frac{3\pi}{2} \) for different values of \( a \). The tangent line equation is \( \mathbf{r}(t_0) + s \mathbf{r}'(t_0) \), where \( s \) is a parameter.

Step 3: Evaluate and Plot for Different a Values

Calculate \( \mathbf{r}(\frac{3\pi}{2}) \) and \( \mathbf{r}'(\frac{3\pi}{2}) \) for \( a = 1, 2, 4, 6 \). Plot the helix and its respective tangent for each \( a \) over \( 0 \leq t \leq 4\pi \) using a computer algebra system (CAS) or a graphing tool.

Step 4: Analyze the Helix as a Varying Parameter a

Observe the changes in the helix and tangents. As \( a \) increases, the helix coils more tightly around the z-axis due to the increased frequency of the trigonometric functions, yielding a more frequent revolution pattern. The tangent line's orientation changes correspondingly, becoming steeper in the xy-plane, indicating how the curve twists more sharply.

Key Concepts

Vector Functions

Vector functions are a powerful tool in mathematics, especially in the study of curves and motions in space. Essentially, a vector function defines a curve in space by mapping a parameter, often denoted as \( t \), to a vector in three-dimensional space.

In the context of a helix, the vector function \( \mathbf{r}(t) = (\cos at) \mathbf{i} + (\sin at) \mathbf{j} + bt \mathbf{k} \) creates a spiral path as it moves through space. The components \( \cos(at) \) and \( \sin(at) \) plot the circular path in the \( xy \)-plane, while the term \( bt \) introduces a vertical rise along the \( z \)-axis.

Some key points about vector functions:

• The vector function provides a position vector for each value of \( t \), giving insight into the point's location at the parameter \( t \).
• It simplifies plotting complex shapes such as helices by breaking down the curve into manageable components.
• By adjusting constants like \( a \) in vector functions, we can observe different behaviors or shapes of the curves.

Tangent Line

The tangent line to a curve is a straight line that just "touches" the curve at a particular point without crossing it, quite like the support of a bookshelf.

In the case of the helix \( \mathbf{r}(t) = (\cos at) \mathbf{i} + (\sin at) \mathbf{j} + bt \mathbf{k} \), the tangent at \( t = \frac{3\pi}{2} \) can be found by deriving the vector function. The derivative \( \mathbf{r}'(t) = (-a \sin at) \mathbf{i} + (a \cos at) \mathbf{j} + \mathbf{k} \) provides the vector direction of the tangent line.

To express the equation of the tangent line, combine the position vector at \( t \) with a parameter \( s \) that scales the direction vector:

• The tangent line is \( \mathbf{r}(t_0) + s \mathbf{r}'(t_0) \), where \( t_0 \) is the given point, and \( s \) is a real number.
• As \( s \) changes, it determines how far along the line you'd like to move in either direction from the tangency point.

This equation shows the tangent's role in approximating the curve's direction around a small interval, similar to predicting a car's path based on its steering at an instance.

Parametric Curves

Parametric curves offer a versatile framework for representing geometrical objects and motions. By using parameters like \( t \), they allow us to express curves that might be challenging to describe with traditional Cartesian equations.

Typically, in the helix example, \( t \) is the parameter controlling the position along the curve, while the expressions for \( x \), \( y \), and \( z \) are defined as functions of \( t \).

Advantages of using parametric curves are:

• They allow separate expressions for each coordinate, making it easy to handle complex shapes and motions.
• They adapt well to varying dynamics; for example, changing \( a \) in the helix modifies the frequency of its twists.
• They also support plotting dynamic scenarios using software tools to visualize the progression over time or through transformations.

In studying a curve like a helix, parametric representation helps in mapping intricate structures and better understanding their attributes.

Computer Algebra Systems

Computer Algebra Systems (CAS) are valuable tools in visualizing and solving complex mathematical problems. They handle symbolic and numerical calculations, providing a versatile platform for exploring concepts like helices in vector calculus.

When plotting a helix and its tangent line over a specified interval, a CAS gives a way to:

• Perform complicated derivative and integral operations with ease.
• Generate precise graphs to visually understand changes in parameter-driven equations.
• Simulate and compare the effects of altering constants like \( a \) and \( b \), offering visual insights into mathematical alterations.

The advantage of using a CAS extends beyond simple computation, making it an educational playground where students can experiment with variables effortlessly and gain better intuition for calculus concepts.