Suppose you leave home and hike 10 miles due north, then 8 miles in the direction 40° east of north,
and then 6 miles due east.
(a) How far are you from home?
(b) What direction should you hike in
order to return home?
same vector stuff like pulling the car
east = 0 + 8 sin 40 + 6
north = 10 +8 cos 40 + 0
dist = sqrt(east^2 + north^2)
tan angle east of N = east/north
(note this is true compass rose angle, not math angle from x axis)
add or subtract 180 depending on if angle was less or more than 180
To calculate the distance from home and the direction you need to hike to return home, we can use vector addition and trigonometry.
(a) To find the distance from home, we can break down the displacements into their north and east components.
For the north displacement:
- The first leg of the hike is 10 miles due north, so the north component for this displacement is +10 miles.
- The second leg of the hike is 8 miles in the direction 40° east of north. To find the north component, we need to use trigonometry. The north component can be calculated as: 8 * cos(40°).
- The third leg of the hike is 6 miles due east, which doesn't contribute to the north component.
So the overall north displacement is 10 + 8 * cos(40°).
For the east displacement:
- The third leg of the hike is 6 miles due east, so the east component for this displacement is +6 miles.
- The second leg of the hike is 8 miles in the direction 40° east of north. To find the east component, we need to use trigonometry. The east component can be calculated as: 8 * sin(40°).
So the overall east displacement is 6 + 8 * sin(40°).
We can now use the Pythagorean theorem to find the distance from home:
Distance = Square root of [(Overall north displacement)^2 + (Overall east displacement)^2].
(b) To find the direction you need to hike in order to return home, we can use trigonometry. The direction can be calculated as the angle of the overall east displacement with respect to due north.
Now, let's compute these values step by step to find the answers.