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Chapter 11: Problem 33

Graph the curves described by the following functions, indicating the positive orientation. $$\mathbf{r}(t)=t \cos t \mathbf{i}+t \sin t \mathbf{j}+t \mathbf{k}, \text { for } 0 \leq t \leq 6 \pi$$

Short Answer

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Question: Graph the curve described by the vector function $$\mathbf{r}(t)=t \cos t \mathbf{i}+t \sin t \mathbf{j}+t \mathbf{k}$$ for $$0 \leq t \leq 6 \pi$$ and indicate the positive orientation. Answer: To graph the curve, plot the point (x(t), y(t), z(t)) for each value of t in the range $$0 \leq t \leq 6 \pi$$, where x(t) = t*cos(t), y(t) = t*sin(t), and z(t) = t. Show the positive orientation by drawing an arrow on the curve starting at the origin (0, 0, 0) and moving in the direction of increasing t.

Step by step solution

01

Understand the vector function components

The given vector function is $$\mathbf{r}(t)=t \cos t \mathbf{i}+t \sin t \mathbf{j}+t \mathbf{k}$$. This defines the position of a point in space in terms of x, y, and z components at a given time t. The components are: - x(t) = t*cos(t) - y(t) = t*sin(t) - z(t) = t These describe how the point moves in the x, y, and z directions as the parameter t changes.
02

Determine the limits of the parameter t

The exercise specifies that the curve should be graphed for the parameter $$0 \leq t \leq 6 \pi$$. This means that we'll evaluate the vector function for t values in the range of 0 to 6蟺.
03

Graph the curve based on the vector function components and the limits of t

To graph the curve, we set up a coordinate system with x, y, and z axes. For each value of t in the range $$0 \leq t \leq 6 \pi$$, we calculate the x, y, and z components using the expressions x(t) = t*cos(t), y(t) = t*sin(t), and z(t) = t. Then, plot the point (x(t), y(t), z(t)) on the coordinate system. You can use computer software or increments of t to plot points along the curve.
04

Indicate the positive orientation on the graph

Finally, we want to show the positive orientation of the curve, which is the direction in which the parameter t increases. To do this, we can draw an arrow on the curve, starting at the point when t = 0 and moving in the direction that the curve goes as t increases. In this case, when t = 0, the curve will start at the origin (0, 0, 0) and move in the direction of increasing t through the coordinate system. By following these steps, you can graph the curve described by the given vector function and indicate its positive orientation.

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Most popular questions from this chapter

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