Q. 12 - Calculus

Question

Let $r\left(t\right)=\left(x\left(t\right), y\left(t\right), z\left(t\right)\right), t\in [a,\infty ),$ be a vector-valued function, where a is a real number. Explain why the graph of r may or may not be contained in some sphere centered at the origin. (Hint: Consider the functions $r_1\left(t\right)=\left(\cos t, \sin t, 1/t\right)$ and $r_2\left(t\right)=\left(\cos t, \sin t, t\right), \mathrm{both} \mathrm{with} \mathrm{domain} [1,\infty ).)$

Step-by-Step Solution

Answer

The graph of r may be contained in a sphere centered at the origin.  

Step 1. Given Information.

The given vector-valued function is $r\left(t\right)=\left(x\left(t\right), y\left(t\right), z\left(t\right)\right), t\in [a,\infty ),$ where a is a real number. 

Step 2. Explanation.

To explain the graph of r may or may not be contained in a circle centered at the origin, let $r_1\left(t\right)=\left(\cos t, \sin t, \frac{1}{t}\right) \mathrm{with} \mathrm{domain} [1,\infty ).$

Now, the graph of $r_1\left(t\right)$ is

Step 3. Explanation.

Now, let  $r_2\left(t\right)=\left(\cos t, \sin t,t\right) \mathrm{with} \mathrm{domain} [1,\infty ).$

The graph of $r_2\left(t\right)$ is

From both graphs, we can depict r that may be contained in a sphere centered at the origin.