b. yVec V(aVec)2 +bVec2, where the superscript T indicates the transpose operation. ZVec=log 10 (avec .. Recall the differentiation between the commands log and log10 c. 6. Matrix operations. Using the variables created in Problems 2 and 3, solve the equations below using matrix operators. (the dots in the below expressions indicate matrix multiplication) b. yMat-(aVec bVec) c.zMat= (bvec, avec) Are yMatandcMat equal? STOP... Before you publish, run your script and make sure that it works! Once you have it running, make sure that you have remembered to not to suppress the outputs with a semicolon. Save your script file and publish your script to a pdf. Remember that if you have not yet used the publish functionality in Matlab, it is likely set to publish in HTML format. Be sure you change the preferred file type to PDF prior to clicking publish. For Problems 7 and 8, create a live-editor file titled "MLXproblem set IB lastname. mix". 7. Engineering Application 1 In 2004, Burt Rutan won the Space X prize with the experimental spacecraft, Space Ship One. The spacecraft was launched at 7,620m (25,000-ft) altitude, climbed to an altitude of 100-km (62 miles), and glided back to land on the same runway from which it and its mother-ship departed. Starting at an altitude (h of 7,620 m, what is the spacecraft's initial vertical velocity (u) required to reach the desired space altitude (h2 of 100,000 m? Provide the answer in units of meters/sec (m/s), feet/sec (ft/s), miles per hour (mph), kilometers/hour (km/h), and knots (kts) by creating five variables: ul mpsec, ul fps, ul mph, ul kmph ul_kts, respectively. Use the convvel command from the aerospace toolbox to convert between units Helpful hints: This problem makes use of the law of conservation of energy, which essentially states that the sum of the potential and kinetic energies of an object at different altitudes must be equal. In equation form, this is represented as where m is the mass of the spacecraft (kg), h is its altitude (m), g is the acceleration due to gravity (9.81 m/s2), and u is the spacecraft's vertical velocity (m/s) If we assume that the spacecraft has expended all of its fuel by the time it achieves its initial vertical velocity (u), then the mass of the spacecraft does not change between altitudes1 and 2. If that is the case, do we really need to know the mass of the spacecraft? Does it matter? If this is not intuitive, solve this problem using m- 1 kg, 1,000 kg, and 10,000 kg and see if the velocity values change What must the vertical velocity (u2) of the spacecraft be when it reaches the maximum altitude (h2 100,000 m)? 8. Engineering Application 2 Plot the following functions for a time vector for a time vector, t, within the range, 0 t 10r. Use an interval of pi/24. Remember, /(t) means fas a function of. Not the f and t multiplied f()-4e5n (3) f(t)-0.3sin (2t) Recall from classroom discussions that this requires the use of"elementwise" or "array" operators",. rather than the "matrix" operators * ^. Plot the three functions in the same figure. For the fi(t), use a solid line For the f (t), use a dash line (-) For the f3(t), use a dash-dot line (). Ensure the box and grid are on (box on and grid on) Label the axes using the xlabel and ylabel commands. Title the figure as "AERN 15300, Prob Set 1, No. 8', using the title command. Create a legend using the legend command to identify the curves. Use the xlim command to set the x-axis limits between 0 and 10 Use the ylim command to set the y-axis limits between-4 and 4 General Instructions: MatlabB organizes data in vectors or matrices, so this problem set is designed to increase your familiarity with using the linear construct of Matlab operations. You must download and use the script template for all your codes, both in the .m and .m1x formats. A 15% penalty will be enforced for not following the prescribed format. Upon completion, students must publish their results as PDF files. M-file (m) scripts are published as pdfs through the MatlabB editor. Live-Editor (.mlx) scripts are published as pdfs using the print button and selecting the "PDF" for the printer Students will submit their assignment online via Blackboard. Make sure that you have fully submitted the correct file by the deadline listed above. Late assignments will be accepted, but as you will see in the Syllabus, come with a substantial reduction in grade. Problems: For Problems 1 thru 6, create a script in m format, titled "problem_set_1A lastname.m" and put all commands in it. Replace the placeholder lastname with your last name. NOT SUPPRESS O THE SEMICOLON IN THIS 1. Scalar variables. Create the following variables: a. a 845 b. b-3.59x10* Do not type in 3 10 8. Remember your scientific notation... c. c-3+5i where i is the imaginary number d. d e where j is the imaginary number and e is Euler's number (use exp, pi) 2. Vector variables. Create the following variables: a. avec =[3 233333-567 12] 8.34 -12 b bVec 1024 2.4 c. cVec 13 2 ..-2 -3(all the numbers from 3 to -3 in increments of 1.0) d. dVec[10 1001 ...109 10(Lorithmically spaced numbers between and 10. Use logspace and make sure the length of the vector is correct (there should be 11 elements). Hirnt logspace is very similar to linspace only in logarithmic fashion rather than linear fashion.) e. eVecHello (this is a string variable, which is a vector of characters) 3. Matrix variables. Create the following variables: 0 23.3333 0 0 0 a. aMat 5.67 0 Note this is a diagonal matrix with the values from Problem 2.a. above on the diagonal. Use the diag command for this exercise. b. bMatsuch that this is a 10x10 matrix with all elements being the number 5. Use either the ones or zeros commands for this problem. cMat such that the diagonal of the matrix is the vector cVec from Problem 2.c. with the number 1 everywhere else within the matrix. This requires the use of the diag and ones commands. Hint: Subtract 1 from each element of cVec, make it diagonal using the diag command, and then add to a ones matrix. c. 8 -6 5 d. dMat2024 2 e. eMat such that it is a 6x6 matrix of random integers between 23 and 64. Use rand, and round floor or ceil commands. 4. Scalar operations. Using the variables created in Problem 1, calculate the variables xl, x2, and x3. Careful with your parentheses and order of operations a.xl b. x2-Va + Vb ) Yes, that is the 21st root of b and the entire expression is raised to the power RI(c+d) (c-d)l sin c. x3- STOP.. Read this carefully before you proceed where R indicates the real part of the complex number in square brackets, c is the complex conjugate of c, and "In" is the natural logarithm or "log base e". Use the real, conj, and log commands for this problem. The Matlab Help Menu is your friend! 5. Vector operations. Using variables created in Problem 2, solve the equations below, elementwise Recall from classroom discussions that this requires the use of "elementwise" operators.,J. -cVec2 Vasy 2n(2.5)