One day you go hiking at a nearby nature preserve. At first, you follow the stra

Question

One day you go hiking at a nearby nature preserve. At first, you follow the straight, clearly marked trails. From your starting point, you travel 2.00 miles down the 1st trail. Then you turn to your left by 30.0° to follow a 2nd trail for 1.10 miles. Next, you turn to your right by 160° and follow a 3rd trail for 1.70 miles. At this point you are getting very tired and would like to get back as quickly as possible, but all of the available trails seem to lead you deeper into the woods. You would like to take a shortcut directly through the woods (ignoring the trails). How far to your right should you turn, and how far do you have to walk, to go directly back to your starting point? Feel free to use the provided vector drawing board to help visualize your work.

Explanation / Answer

Here ,

let the initial direction is x axis

d1 = 2 i

d2 = 1.1 * (cos(30) i + j * sin(30)) mile

d2 = 0.95 i + 0.55 j miles

d3 = 1.7 * (cos(130) i - j * sin(130)) miles

d3 = -1.09 i - 1.30 j miles

for the net displacement

dnet = d1 + d2 + d3

dnet = 2 i + 0.95 i + 0.55 j - 1.09 i - 1.30 j

dnet = 1.86 i -0.75 j

dnet = sqrt(1.86^2 + 0.75^2) at arctan(-0.75/1.86) degree

dnet = 2 miles at - 22 degree

he should turn left = 130 -22 = 108 degree

he should travel 2 miles after turning 108 degree left

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