You are analysing data collected by a rover on the moon. You mayassume that the rover starts at the origin (0,0,0), and that eachcoordinate it transmits contains an East-West component, aNorth-South component and an altitude component, relative to theorigin. Let positive movement in the i directionbe East, positive movement in the j direction beNorth, and positive movement in the k direction beincreasing altitude. All values are in meters. Assume the moon isflat (neglect its curvature). The rover relays the following fourcoordinates in order, as it descends into a crater. Coordinates areissued each time the rover changes direction. As such, you mayassume that each trajectory between these points is a straightline:
O : (0, 0, 0).
P1: (2000, 5000, -500).
P2: (3000, 8000, -600).
P3: (6000, 9000, z).
Unfortunately, the transmission of the fourth coordinate iscorrupted, and does not have an altitude component. For now, youlabel this component as z.
(a) Determine the displacement vectorsdescribing the rover's straight-line trajectory between each oftheir transmissions. You may keep z as an unknown.
(b) Upon reaching P2, the rover'sbattery has drained and requires recharging. Assuming the batterywas initially at full charge, calculate the approximate range ofthe rover from one battery charge (i.e. what distance can it travelbefore requiring a recharge).
(c) For communication with the rover, missioncontrol needs to point their antenna (located at O)towards the rover. Ignoring the altitude component (i.e. treat theproblem as two-dimensional), calculate the angle that the antennamust be rotated to point towards P2 if it is currentlypointing at P1.
(d) Based on the time taken to send and receivecommunications, you calculate that at P3 the rover mustbe 10,850 meters from O. Determine z, thealtitude component of the co-ordinate P3. You may assumethat P3 is the lowest point in the crater.
