Imagine you've been assigned to work on the operations team for the Mars rover, Curiosity. The landing site is located at $(0,0)$. Since landing, the rover has moved in a straight line to a spot $A$ that is 3 km west and 9 km south of the landing site. The science team wants the rover to reach a location $B$ that is 16 km west and 10 km south of the landing site.
- Express the rover's current position $A$ as a vector relative to the landing site. How far is the rover from the landing site? Give your answer exactly (in terms of square roots) rather than converting to decimals.
- Express the desired destination $B$ as a vector relative to the landing site. Then give a vector $\vec{AB}$ which points from the current location to the desired destination. What vector operation did you use?
- How far must Curiosity travel to reach the new destination? Give a unit vector $\hat{AB}$ that points from the current location to the desired location.
- You decide to have the rover move in a straight line from $A$ to $B$. Since the rover has been moving in the direction of $\vec A$, you first need to tell the rover to rotate clockwise till it points in the direction of $\vec {AB}$. How many degrees should the rover turn? (Use a calculator to simplify.)
- The rover has enough power from solar panels to move about 2 km a day. Give a vector that represents the displacement the rover makes in one day while moving in the direction of the vector $\vec {AB}$.
- Give the $(x,y)$ location of the rover after 1 day. Don't worry about simplifying your answer. Then give the $(x,y)$ location of the rover after 2 days. The goal is to notice a pattern so you can write a formula for any number of days.
- Give the $(x,y)$ location of the rover after $t$ days.
- How does your answer to the previous change if the rover can move 3 km per day?
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