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Chapter 11: Problem 32

Graph the curves described by the following functions, indicating the positive orientation. $$\mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+2 \mathbf{k}, \text { for } 0 \leq t \leq 2 \pi$$

Short Answer

Expert verified
Question: Graph the curve described by the function $$\mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+2 \mathbf{k}, \text { for } 0 \leq t \leq 2 \pi$$ and indicate its orientation. Answer: The curve is a circle with a radius of 2 units, centered on the z-axis at a height of z = 2, and oriented counterclockwise when viewed from above.

Step by step solution

01

(Determinate the Components)

First, we will determinate the components functions based on the given function \(\mathbf{r}(t)\). We have: $$x(t) = 2\cos t,$$ $$y(t) = 2\sin t,$$ $$z(t) = 2.$$ Using these functions, we can plot the given curve.
02

(Creating a table of values)

Next, we can create a table of values for \(x(t)\), \(y(t)\), and \(z(t)\) at specific points within the given interval, \(0 \leq t \leq 2\pi\). Here's a table with some values: | \(t\) | \(x(t)\) | \(y(t)\) | \(z(t)\) | |------------|---------|---------|---------| | \(0\) | \(2\) | \(0\) | \(2\) | | \(\frac{\pi}{2}\) | \(0\) | \(2\) | \(2\) | | \(\pi\) | \(-2\) | \(0\) | \(2\) | | \(\frac{3\pi}{2}\) | \(0\) | \(-2\) | \(2\) | | \(2\pi\) | \(2\) | \(0\) | \(2\) |
03

(Plotting the points)

Now that we have our table of values, we can plot the points on a 3D coordinate axis. The points are as follows: 1. \((2, 0, 2)\) 2. \((0, 2, 2)\) 3. \((-2, 0, 2)\) 4. \((0, -2, 2)\) 5. \((2, 0, 2)\)
04

(Graphing the curve)

Connect the points as they appear in the table to form the curve. Our graph should look like a circle in the \(x-y\) plane at a height of \(z = 2\). Because the table moves from the positive \(x\)-axis to the positive \(y\)-axis, this defines the positive orientation of the curve.
05

(Summing up the curve)

In conclusion, the curve described by the function $$\mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+2 \mathbf{k}, \text { for } 0 \leq t \leq 2 \pi$$ is a circle with a radius of \(2\) units, centered on the \(z\)-axis at a height of \(z = 2\), and oriented counterclockwise when viewed from above.

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Most popular questions from this chapter

Find the function \(\mathbf{r}\) that satisfies the given conditions. $$\mathbf{r}^{\prime}(t)=\left\langle e^{2 t}, 1-2 e^{-t}, 1-2 e^{t}\right\rangle ; \mathbf{r}(0)=\langle 1,1,1\rangle$$

Evaluate the following definite integrals. $$\int_{-1}^{1}\left(\mathbf{i}+t \mathbf{j}+3 t^{2} \mathbf{k}\right) d t$$

Relationship between \(\mathbf{r}\) and \(\mathbf{r}^{\prime}\) Consider the circle \(\mathbf{r}(t)=\langle a \cos t, a \sin t\rangle,\) for \(0 \leq t \leq 2 \pi\) where \(a\) is a positive real number. Compute \(\mathbf{r}^{\prime}\) and show that it is orthogonal to \(\mathbf{r}\) for all \(t\)

Use the formula in Exercise 79 to find the (least) distance between the given point \(Q\) and line \(\mathbf{r}\). $$Q(6,6,7), \mathbf{r}(t)=\langle 3 t,-3 t, 4\rangle$$

Compute the indefinite integral of the following functions. $$\mathbf{r}(t)=t e^{t} \mathbf{i}+t \sin t^{2} \mathbf{j}-\frac{2 t}{\sqrt{t^{2}+4}} \mathbf{k}$$
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