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}]Mathematica Reference Sheet Winter 2023 Save Early 34 Save Often! When using Mathematica it is important to remember to save frequently to avoid losing your work! Mathematics Home Use Licenses: SU students now have access to a free Mathematics HomeUse license. Please see the "Student HomeUse Mathematica License" page on Canvas for more information. ccessing Mathematica on campus computers: You should be able to access Mathematics in any of the computer labs on campus. You can also access Mathematica on your personal computer using the Redhawk Labs. Redhawk Labs Link: https:ffseattleu.apporto.comf Redhawk Labs User Guide: https:ffseattleuniversity.atlassiaIL.netfservicedeskfcustomer! port aler'E-Iticlevf1224048641?src=1530904659 Top 4 Rules of Mathematica: Keep the Following rules in mind when writing Mathematica code: 1. Capital letters on all cemmand names 2. [ ] surround function [or command} arguments 3. { } are used for lists and ranges 4. Shift l Return to evaluate input For example, if we wanted to plot 3,! = 3:3 from ft? = 3 to :r: = 3 and label the axes, our code would look like: Plot[x"3, {1, -3, 3}, llxeSLabel -> {1c, 3r}] Mathematica Cells and Word-ProceSsing Capabilities: 1. Mathematica documents are divided into cells. The software has its own builtin word processor. You can then intersperse text with active Mathematica commands and the output from these commands: formulas, solutions, and graphics. 2. Mathematica cells are delineated by brackets on the right side of the window. The brackets do not appear in the printed version. New cells are automatically created as Input cells. To choose a different type of cell, highlight the bracket on the right side of the window, then select Style from the Format pulldown menu. You can change fonts, add bold or italic text, etc. To start a new cell, click the mouse below your current cell, and a (temporary) horizontal line appears. Text and commands cannot be mixed in the same cell. 3. Formatting your Ma thematica notebooks so that they are separated into sections and subsections is often desirable. To divide your notebook into sections, create a new cell, highlight the bracket on the right side of the window, and select "Section" lmm the Format > Style pulldown menu. 1of2 Mathematics Facts: The list below contains some helpful Facts about working with Mathematica. Additional help can be found in the software help features, including the Index of Functionsr in the Documentation Center. 1. 13. 11. 12. All Mathematica commands begin with capital [upper case} letters. Mathematica is case sensitive. Therefore "A" and "a" are difFercnt variable names, as are spellings of names with changes in case. Care is needed when typing. . Mathematica commands use square brackets [ ] to enclose their arguments {values you type in). Ex amples: Exp[x] , Sin [xsy] , Log [3:] , Cos [3t] , Tan [x] , Eqrt [3t]. . Round parentheses L} and curly brackets {,lr are used only for grouping and listing. . To multiply variables, always use either the asterisk C") symbol or a space. Typing "xy" will not be interpreted as multiplying x and y. It will be interpreted as a single variable named "xy." Typing x=ty or at y will be interpreted as multiplying. . The capital letters C, D, E, I, III, and D are reserved for Mathematica. Do not use them as variable DENIES. . Use the"Enter" key or "Shift l Return" to tell Mathematica to execute a Command. The "Return" key alone will not execute a command, but will just give a new line while typing. . To evaluate an entire notebook, in order from top to bottom, you can select "Evaluate Note book" From the Evaluation pulldown menu. (This is what I do when I grade your submitted Mathematica files.) . Remember to save your work often. Mathematica does not automatically save a copy of your work. . If Mathematica starts behaving strangely, you can wipe the current session's memory by se lecting "Quit Kernel >Local" from the Evaluation pulldown menu. To get information about a Specific command, type a "?" before the command in an input cell and execute it. For example, executing 'E'Exp will result in a short description of the "Exp" command, with a link that you can click on for more information in the Documentation center. Palettes: The pulldown menu Palettes has various items. Each will give a window containing useful selections. You will probably want to use Classroomissistant palette. Freeform Input: If you are working on a computer with an internet connection, you may type "=" at the beginning of a cell to switch to freeform input mode. This may be useful if you want Matheniatica to do something, but you don't know the name of the command to use. You can type instructions in plain English, and have Mathematics translate them into commands. Note that when you turn in your Mathematica labs, any "freeform input" should be replaced with Mathematica commands