91Ó°ÊÓ

Use a computer algebra system to graph the vector-valued function \(\mathbf{r}(t) .\) For each \(\mathbf{u}(t)\) make a conjecture about the transformation (if any) of the graph of \(\mathbf{r}(t) .\) Use a computer algebra system to verify your conjecture.\(\mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+\frac{1}{2} t \mathbf{k}\) (a) \(\mathbf{u}(t)=2(\cos t-1) \mathbf{i}+2 \sin t \mathbf{j}+\frac{1}{2} t \mathbf{k}\) (b) \(\mathbf{u}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+2 t \mathbf{k}\) (c) \(\mathbf{u}(t)=2 \cos (-t) \mathbf{i}+2 \sin (-t) \mathbf{j}+\frac{1}{2}(-t) \mathbf{k}\) (d) \(\mathbf{u}(t)=\frac{1}{2} t \mathbf{i}+2 \sin t \mathbf{j}+2 \cos t \mathbf{k}\) (e) \(\mathbf{u}(t)=6 \cos t \mathbf{i}+6 \sin t \mathbf{j}+\frac{1}{2} t \mathbf{k}\)

Step by step solution

01

Graph the Original Function

First, use a computer algebra system to graph the vector-valued function \( \mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+\frac{1}{2} t \mathbf{k}\).\n This will form the basis for comparing all transformations.

02

Graph Function (a)

\(\mathbf{u}(t)=2(\cos t-1) \mathbf{i}+2 \sin t \mathbf{j}+\frac{1}{2} t\mathbf{k}\) changes \(\mathbf{r}(t)\) by subtracting 1 from \(\cos t\), which shifts the graph 1 unit left along the x-axis. Graph the given function and compare the result with the graph of the original function to verify this speculation.

03

Graph Function (b)

For the function \(\mathbf{u}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+2 t \mathbf{k}\), the k-component of \( \mathbf{r}(t)\) is multiplied by 2, which is a scalar multiplication of the z-axis. Therefore, the plot will stretch in the z-direction. This hypothesis can be checked by graphing the function.

04

Graph Function (c)

The function \(\mathbf{u}(t)=2 \cos (-t) \mathbf{i}+2 \sin (-t)\mathbf{j}+\frac{1}{2}(-t) \mathbf{k}\) changes \( \mathbf{r}(t)\) by replacing \(t\) with \(-t\) . This will cause the graph to be reflected in the xy-plane and the direction of the curve to be reversed. Validate this conjecture by charting the given function.

05

Graph Function (d)

In the case of \( \mathbf{u}(t)=\frac{1}{2} t \mathbf{i}+2 \sin t \mathbf{j}+2 \cos t\mathbf{k}\), \( \mathbf{r}(t)\) is rearranged. The x-component now progresses linearly with time, the z-component draws the cosine graph, and the y-component creates the sine graph. Graph this function to observe the transformation.

06

Graph Function (e)

Lastly, \(\mathbf{u}(t)=6 \cos t \mathbf{i}+6 \sin t \mathbf{j}+\frac{1}{2} t\mathbf{k}\) modifies \( \mathbf{r}(t)\) by multiplying the x and y components by 3, which scales the graph in the xy-plane by a factor of 3. Graph this function to confirm this transformation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transformation

When we talk about transformations in vector-valued functions, we refer to changes in the geometric representation of a function. These modifications can include translations, rotations, reflections, or scalings. Understanding transformations helps in predicting how a graph will change in shape or position when manipulated.
In our given vector-valued function \( \mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+\frac{1}{2} t \mathbf{k} \), various transformations were applied to create new functions \( \mathbf{u}(t) \).

• For instance, in function (a), subtracting 1 from \( \cos t \) translates the graph along the x-axis.
• Function (c) involves a reflection due to the replacement of \( t \) with \( -t \), effectively reversing the direction.
• In function (e), multiplying the x and y components by 3 scales the graph in the xy-plane.

Comprehending these transformations allows you to predict and verify graph changes, which is crucial for applications in physics and engineering.

Graphing

Graphing vector-valued functions is a way of visualizing the paths traced by these vectors as the parameter \( t \) varies. Each function \( \mathbf{r}(t) \) or \( \mathbf{u}(t) \) describes a curve in three-dimensional space. By graphing, you can examine how different components—such as i, j, and k—contribute to the trajectory of the path.
Initially, you graph \( \mathbf{r}(t) \) to establish a baseline for comparing other modified functions. This step is crucial because it visually lays out the effects of any transformations.

• Adding or subtracting values to components shifts the graph along respective axes.
• Multiplying a component stretches the graph, and reflects it across an axis by changing the sign of a parameter.

By using graphing to explore these aspects, you deepen your understanding of not only the function itself but also the roles each component plays in defining the path.

Computer Algebra System

A Computer Algebra System (CAS) is a software tool used to perform symbolic mathematics. It handles algebraic equations, calculus operations, and graphing functions, which makes tasks involving vector-valued functions much smoother and more intuitive.
Using a CAS for our functions \( \mathbf{r}(t) \) and \( \mathbf{u}(t) \), you can effortlessly plot graphs to observe transformations and validate geometric conjectures.
With the ability to instantly alter function parameters and observe outcomes, a CAS facilitates:

• quick verification of mathematical hypotheses,
• accuracy in computations and graphical plotting,
• an educational aid to deeply understand the graphical transformations.

Engaging with a CAS improves learning efficiency by providing a hands-on experience of mathematical concepts, essential for students tackling complex problems.

Scalar Multiplication

Scalar multiplication in vector-valued functions involves scaling a vector by a constant factor. This operation impacts the magnitude of the vector without altering its direction.
Let's consider function (b) : \( \mathbf{u}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+2 t \mathbf{k} \). Here, multiplying the k-component by 2 results in a doubling of the z-axis component of the vector. Accordingly, the graph extends further along the z-axis, effectively stretching the original curve.
Similarly, function (e) modifies both the x and y components by multiplying them by 3, enlarging the scale within the xy-plane.

• Scale adjustments, through scalar multiplication, allow for changes in size, rather than shape.
• This method is useful in modeling real-world scenarios where proportionality is key.

Understanding scalar multiplication enhances your ability to control vector magnitudes in mathematical and physical applications.