Activity 9.2.3.
In this problem, we will be navigating to help find a friend who is lost at a local state park. We will be navigating using traditional map coordinates with east going in the positive \(x\)-direction and north going in the positive \(y\)-direction.
(a)
Your friend told you they would be staying 3 km east and 4 km north of the main parking lot. You drive to the parking lot and park next to your friend’s car, then hike 3 km east and 4 km north. You get to the location you expected your friend to be at, but you don’t find your friend and call the ranger station. The ranger station says they think your friend is 1 km west and 4 km north of your current location.
Let \(\vu\) be the vector that represents your hike from your car to the expected location of your friend, \(\vv\) be the vector that represents the vector from your current location to the ranger’s suggested location, and \(\vw\) be the vector from your car to the ranger’s suggested location. Compute the components of \(\vu\text{,}\) \(\vv\text{,}\) and \(\vw\text{.}\)
(b)
Draw a picture that represents \(\vu\text{,}\) \(\vv\text{,}\) and \(\vw\) with the reference points \(C\) being the location of your car, \(P\) being your current position, and \(S\) being the ranger’s suggested location.
(c)
In the context of this problem, explain why you can add the horizontal components of \(\vu\) and \(\vv\) to get the horizontal component of \(\vw\text{.}\) Write a sentence or two about why this argument should work for the vertical components as well.
(d)
After hiking to the location suggested by the ranger, you still don’t see your friend and call the ranger station again. The regional manager of rangers answers this time. She says the first ranger made a mistake in their navigation. A drone spotted your friend along the same direction from your car to point \(P\text{,}\) but your friend is three times as far away from your car as you were when you were at point \(P\text{.}\) In order to avoid confusion or any other mistakes, you want to compute the vector from your car to the drone’s suggested location, which we will call \(D\text{.}\) What are the components of \(\overrightarrow{CD}\text{?}\) How does that compare to \(\vw\text{?}\)
(e)
After hiking to the drone’s suggested location, you find your friend. Yay! On the walk back to your car with your friend, you decide to help make sure the first ranger understands vectors. In particular, you want to let them know what direction they should have told you to go from \(P\text{.}\) Use the components of \(\overrightarrow{CD}\) and \(\vu\) to figure out what the first ranger should have told you.
