Question 2.4 (13 marks) Now that you have set up your navigation aircraft in the previous question, you can commence your actual mission: launching an interstellar spaceship through a wormhole! A wormhole is a strange 4- dimensional place, but your engineers were able to map it out with a coordinate system. The spaceship is currently in position (0, 0, 0, 0) and it has to reach position (8, 29, 57, 74) in order to achieve its mission. The spaceship has six jets that allow it to go up, down, left, right, forward, backward. In the wormhole, however, these directions don't exist anymore! Instead, moving one unit up, down, left, right, forward, and backward now corresponds to moving by u = (1, 2, 3,4), d = (1,2,3,1),l = (2,0,0, 2), r = (3, 5, 3, 3), f = (0,4, 6, 6), andb = (l,1, 1,1), respectively. What amess! a) You need to program the spaceship and decide how many units 15, d, l, r, f, b the spaceship must travel in each of the directions u, d, l, r, f, and b respectively, in order to get from (0, 0,0,0) to (8, 29, 57, 74). Write down a system of four equations with six unknowns that represents this information. Then represent this linear system in its matrix form (i .e. as a matrix equation). b) Using Gaussian elimination, nd all solutions to the equations in a). Write your set of solutions also in vector form. (Tip: test that your solution is correct!) 0) A meteoroid hits the spaceship and damages the forward and backward jets so that you cannot use these anymore. Can the spaceship still go from (0, 0, 0, 0) to (8,29,57,74)? What is the answer if the meteoroid damages the right and backward jets instead? Question 2.3 (8 marks) Three aircraft (called A, B, and C) are positioned in space; the position of each aircraft is specied by three space coordinates (any, 2:), given with respect to the space station having coordinates (0,0,0). Aircraft A has xed position (2, 5, 8), and aircraft B has xed position (3, 5, 8). Write a and b for the 3-dimensional vectors pointing from the space station to A and to B, respectively. Do the following. a) Determine the distance between aircraft A and B. b) You want to place aircraft C such that the distance between B and C is the same as the distance between B and A, and such that the line through B and C is orthogonal to the line through B and A. Determine the set of all possible coordinates (c1, C2, C3) of C. Hint: your answer should be a set of triples (c1, (:2, (:3) that satisfy certain explicit equations