Question

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) \(\mathbf{u}(t)=6 \cos t \mathbf{i}+6 \sin t \mathbf{j}+\frac{1}{2} t \mathbf{k}\)

Short answer

Each transformation of \( \mathbf{u}(t) \) with respect to \(\mathbf{r}(t)\) depends on the particular changes made to \(\mathbf{r}(t)\) to get \(\mathbf{u}(t)\), and can involve scaling, shifting, or reflecting, among others. The verification of the transformations would depend on using a computer algebra system, and would likely show that our earlier conjectures about each transformation are essentially correct.

Step-by-step solution

Determine \(\mathbf{r}(t)\)

For the vector-valued function : \(\mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+\frac{1}{2} t \mathbf{k}\), plot using a computer algebra system to visualize the graph.

Determine \(\mathbf{u}(t)\) a-e and Conjecturing

For each \(\mathbf{u}(t)\), develop a conjecture about its transformation relative to \(\mathbf{r}(t)\) based on the changes observed in \(\mathbf{u}(t)\) compared to \(\mathbf{r}(t)\), and visualize the graph. For example, in the case of (a), it seems that subtracting 1 from the \(\cos t\) factor of the i component could potentially shift the graph left by 1 unit.

Verification

Use the computer algebra system to plot each \(\mathbf{u}(t)\) alongside \(\mathbf{r}(t)\) to verify the conjecture. If the conjecture proves correct, the change and transformation from \(\mathbf{r}(t)\) should be clearly visible in each graph of \(\mathbf{u}(t)\).