Please answer each problem in Mathematica code.

WOLFRAM MATHEMATICA Plan: *23FQ-Math-1336-Projex1).nb File Edit Format Insert Goals of this Project: The goals of this project include: extending the concepts we learned about vectors, lines, and planes to find an angle between a given line and a plane; exploring tangent lines to space curves; and calculating lengths of space curves. Logistics & Instructions: Projects are due by 11:59 PM on Wednesday, November 29, 2023 by 11:59 PM. Projects may be done individually, or in groups up to three students. Groups may consult with one another, but each group must write up their own report based on their own understanding. One member from each group should upload the Project Notebook to the "Mathematica Project" assignment on Canvas. The names of all group members must be included at or near the top of the Lab file. Each student should complete the "Project Group Survey" assignment, which will allow me to set up the groups on Canvas. (Please do this even if you choose to work individually!) Note that there is a discussion forum for this assignment in Canvas. Please make use of this resource to collaborate with your classmates! " All answers/solutions must be explained completely and clearly. " (2 pts. ) Formatting Expectations: Please make sure that all written explanations are formatted as "text" and that each problem is contained in a separate section or subsection. When grading your projects, I will run your code by going to the Evaluation menu and selecting "Evaluate All Cells." Please make sure that you try this and make sure that everything works properly before submitting your project. Problem 1: A Line and a Plane meet an an angle... In this problem, we consider the line with parametric equations: x =7 -5t, y = 2 - 9t, z= - 6-5t, and the plane with equation: 5x - 8y + 2z = 13. (a) (2 pts.) Complete the commands below that will allow you to plot the line and the plane. In[.J.= line = ParametricPlot30[ { , , ), {t, -5, 5), Axeslabel - {x, y, z), ViewPoint - {1, 1, .75}] plane = Plot3D[ , {x, -5, 10}, {y, -5, 5), Axeslabel - {x, y, z), ViewPoint - {1, 1, .75), PlotStyle - Opacity[ .5, Green] ] (b) (2 pts.) The Show[ ] command below allows you to view the line and the plane on the same axes. Use the PlotRange-> and Viewpoint-> options to modify the graph so that you can clearly see the intersection point, and can get an idea of what the angle between the line and plane should be. In[ .)= Show[plane, line] (c) (2 pts.) How can you use the angle between the line (above) and the vector normal to the plan to find the angle between the line and the plane? Explain how to do this in words (a few sentences). (d) (4 pts.) Use Mathematica to carry out the calculations you described in the previous part, and find the angle between the line and the plane. (e) (2 pts.) Does the value that you calculated for the angle between the line and the plane make sense? Why or why not? Be sure to reference the plots you generated. Inof=Problem 2: Space Curve In this problem, we will consider the space curve whose equation is shown below: r (t) = 5sin(2t) i + (sin(t) - t + 4 cos(3t)) j + (3t - 2 cos(t)) k (a) (1 pt.) Define r(t) In [ ] = r [ t_ ] = (, , ) (b) (3 pts.) Plot the space curve for -27t S t S 3/T. Set up your command so that the graph is named "curve," by entering curve = ParametricPlot3D[ ...fill this part in yourself...] (c) (2 pts.) Use Mathematica to find the equation of the line tangent to the curve at t=4, and define the equation as I(t): In[ .J= (d) ( 1 pt.) Plot the tangent line. Set up your command so that the graph is named "tangent," by entering tangent = ParametricPlot3D[( ..fill this part in yourself...)] In.J (e) (2 pts.) Use a Show command to plot the curve and the tangent line on the same axes. Do you see what you expected to see? In[ .]:= (f) ( 2 pts. ) Use an Nintegrate[ ] command to find the length of the curve plotted in part b. Does the value you calculated make sense?Top 4 Rules of Mathematica: Keep the following rules in mind when writing Mathematica code: 1. Capital letters on all command names 2. [ ] surround function (or command) arguments 3. { } are used for lists and ranges 4. Shift | Return to evaluate input For example, if we wanted to plot y = a from r = -3 to r = 3 and label the axes, our code would look like: Plot [x 3, {x, -3, 3), AxesLabel -> {x, y}]