MAT188-WRITTEN-HOMEWORK 1, Sep 21, 11:59 PM Problem 2. After a long journey in the woods, you find yourself in front of a cabin. The cabin belongs to an old man named Gausse. When you enter, Gausse reveals to you that he has been in contact with aliens. The aliens live in a separate universe, however there is a portal through which they have contacted Gausse. Gausse has set a coordinate system, with the origin right on the ground at the centre of his property. Going through his papers, he shows you a description of the plane on which the portal is located. This plane does not intersect our planet earth. x A={y,x+y − z = −100}. Gausse has also built a rocket ship, and invites you to ride out with him on it to find the alien portal on the plane A. Once aimed, the rocket ship cannot change directions. Don't think too hard about how you're going to get home on a one-way rocket, I'm sure Gausse has a plan. (1) The rocket ship is set to be launched from the launchpad, located at the origin. The rocket is Determine if you would encounter the mysterious aimed in the direction of the vector 1 plane on your journey. Support your answer with a complete mathematical argument. (2) Use vector form in proper set builder notation to describe the trajectory of the rocket. Gausse reminds you that his rocket does not go backward and cannot tunnel through the earth. (3) Insert a Geogebra image that shows A, and the trajectory of the rocket. Does the image back up your argument from part? (4) Gausse wants you to suggest a direction for the rocket such that an encounter with the plane A is guaranteed, provided that the rocket is launched from the launchpad. Give such a vector and explain why it works. He promises to program the rocket to be aimed in this direction if he is convinced. (5) After arriving to the plane, Gausse has enough fuel to travel 12 units, what is the furthest point you could get to if you aim in the direction 1. Give the coordinates of this point. (6) You successfully make it to the plane. The aliens, who were prepared for your arrival, have left you a new ship in which you can dock your own. The new ship has the ability to go in the 0 When you board, you receive a radio message that they will wait for directions 1 and 0 15 you at their latest position with position vector 20 135 Can you make it there?