Dos anybody know how I do this on a computer? Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.
r(t) = <sin(3t)cos(t),t/4,sin{3t)sin9t)>
There are lots of online graphing sites. But good luck here.
You know that cos(t),t/4,sin(t) is a helix, right? As x and z trace out their circle, the locus moves along the y-axis. But this is an insanely complicated Lissajous figure in the x-z plane, which is also stretched along y as it is traced.
geogebra.com does 3-D parametric curves. Start out with a simple command, like (sin(t),t,cos(t)) to get a feel for using it.
To graph the curve with the given vector equation on a computer, you can follow these steps:
Step 1: Choose a parameter domain:
Select a suitable range of values for the parameter t that will cover the desired portion of the curve. Since the equation involves trigonometric functions, specifying a domain that covers one complete period will generally be sufficient. In this case, you can choose a domain of t values from 0 to 2π.
Step 2: Create a table of values:
Choose some equally spaced values within the parameter domain and calculate the corresponding x, y, and z coordinates using the vector equation r(t). For example, you can select t values such as 0, π/4, π/2, 3π/4, π, and 2π.
Step 3: Plot the points:
Use a graphing software or a spreadsheet program to create a scatter plot of the calculated points from the previous step. Use the x-coordinate values as the x-axis, y-coordinate values as the y-axis, and z-coordinate values as the z-axis.
Step 4: Connect the points:
To visualize the nature of the curve, connect the plotted points in the order they were calculated. This will give you a continuous curve that represents the shape of the graph.
Step 5: Adjust the viewpoints:
Rotate or zoom in/out of the 3D graph to achieve viewpoints that reveal the true nature of the curve. Different software tools have different navigation features that allow you to adjust the view. Experiment with these controls until you get a clear understanding of the curve's shape.
Note: Specific instructions for using graphing software will depend on the program you are using. Most graphing software, such as Desmos, GeoGebra, or Matlab, provide clear documentation and tutorials on how to input equations, plot curves, and manipulate viewpoints.
To graph the curve with the given vector equation on a computer, you can follow these steps:
1. Choose a software: There are several software options available for graphing curves, such as Matlab, Mathematica, or online tools like Desmos or GeoGebra. Choose the software that you are most comfortable working with or have access to.
2. Input the vector equation: Once you have the software open, you need to input the vector equation into the software. In this case, the vector equation is r(t) = <sin(3t)cos(t), t/4, sin(3t)sin(9t)>. Make sure to properly represent the trigonometric functions (e.g., sin, cos) and use the correct multiplication symbols (*) when necessary.
3. Define the parameter domain: Specify the range of values for the parameter "t" that you want to graph. Usually, you can choose a range that allows you to see multiple cycles of the curve. For example, you can use a parameter domain such as t ∈ [-2π, 2π] to view two full cycles of the curve.
4. Set the viewpoints: Determine the viewpoints or angles from which you want to view the graph. This will help you reveal the true nature of the curve. You can experiment with different viewpoints to get the desired perspective. For instance, you may want to view from different angles, sides, or directions to gain a better understanding of the curve's shape.
5. Plot the graph: With the vector equation, parameter domain, and viewpoints set, you can instruct the software to plot the graph. Simply press the corresponding buttons or run the code to generate the plot.
6. Analyze the graph: Once the plot is generated, examine the graph to gain insights into the nature of the curve. Pay attention to the shape, symmetry, intersections, and any other relevant features that you want to observe or study.
Remember, the specific steps may vary depending on the software you choose to work with, but the general process remains the same.