Question

Use a computer algebra system to graph the vector-valued function and identify the common curve. $$ \mathbf{r}(t)=-\frac{1}{2} t^{2} \mathbf{i}+t \mathbf{j}-\frac{\sqrt{3}}{2} t^{2} \mathbf{k} $$

Short answer

After plotting the given vector-valued function, analyze the graph to identify the form of the curve it forms in the three-dimensional plane.

Step-by-step solution

Input the vector function in a computer algebra system

In order to plot the vector-valued function \(\mathbf{r}(t)=-\frac{1}{2} t^{2} \mathbf{i}+t \mathbf{j}-\frac{\sqrt{3}}{2} t^{2} \mathbf{k}\) in a computer algebra system such as Python or MATLAB, first define or script this function in the platform you are using. In Python, you can use the numpy and matplotlib package to perform this.

Display the 3D plot of the function

After defining the function, the next action is to display the 3D plot of the vector-valued function. Using matplotlib, you can create a 3D figure and then plot the function. Remember to label your axes as i, j, and k or x, y, and z.

Identify the common curve

Once you have plotted the vector function, identify the curve that it is forming. Usually, a vector valued function can form a line, circle, ellipse, parabola or hyperbola in 3D space. Identification should be done based on observation of the plotted function in the 3D space.