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MindBook decides to diversify their portfolio and invest in the "RoboMap" technology which is a robot that maps different types of territories. The new CEO is trying to make some drastic changes to some of the software and hardware systems in the new version of the robot. You have been hired by the opposing faction in the board to lead an investigation on how the performance of the robot is affected by the new proposed changes.

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a$)$ The first system that you will test is the navigation system. The robot maps the territory by calculating distances and angles whilst moving between various locations. It then stores on its system the shortest direct path from initial location to the intended target During the test the robot navigates around a muddy surface in the following path: $55cm[$NW$]$ and then moves $45cm[S30W].$ The robot sends back to the server the following overall shortest path $].$ Use triangle or parallelogram method to verify the precision of the robot's navigation system.

b$)$ The next test involves the performance of the new battery. The robot encounters an obstacle on his journey Using its arm it applies a force of $35N$ at an angle of $25$ below horizontal and is able to displace the object for $20m$ in a time of $65s.$ At the beginning of the test the battery's is displaying a total stored energy $($ability to perform mechanical work$)$ of $2500Nm.$ What would be the remaining energy displayed at the end of the test? Useful formula: $E=W($change in available energy is equal to the work done$)$

c$)$ The robot is also able to lift itself by pushing its arm against the ground in a rotational motion around an axis at the back wheel. This is useful to move on top of obstacles encountered during exploration. The robot applies a force $10,5,15$ at a point where the arm vector is $-5,2,3.$ Determine the torque vector.

d$)$ One exploratory mapping session for a $1m$ by $1m$ can last $5$ minutes. The new mapping system should be able to handle absolute coordinates in space as well. That is the compass directions might not be available. This mapping system uses cartesian system using as origin $(0,0,0)$ the landing site. The robot starts the exploration at the point $(1,2,-3)$ and ends the exploration at the point $(2,0,1).$ Describe the trajectory of the move using the equation of line between the starting and the final location of the robot.