A man was lost in the woods. He wanders 3 km north, then 7 km east, then 7 km south, then 4 km west. What is the magnitude and direction of his resultant displacement?
south 7-3 = 4
east 7-4 = 3
d = sqrt (9+16) = 5km
tan theta = 3/4 where theta is angle east of south
theta = 36.9 deg east of south
or 180 -36.9 = 143.1 on your compass clockwise from north
To find the magnitude and direction of the man's resultant displacement, we can use vector addition.
Let's break down each step of the man's movements into north, east, south, and west components:
1. North: 3 km
The north component is positive, as the man is moving in the northern direction. The vector for this component can be represented as (0, 3).
2. East: 7 km
The east component is positive, as the man is moving in the eastern direction. The vector for this component can be represented as (7, 0).
3. South: 7 km
The south component is negative, as the man is moving in the opposite direction. The vector for this component can be represented as (0, -7).
4. West: 4 km
The west component is negative, as the man is moving in the opposite direction. The vector for this component can be represented as (-4, 0).
To find the resultant displacement, we need to add up all the components:
Resultant displacement = (0 + 7 + 0 - 4, 3 + 0 - 7 + 0)
= (3, -4)
The magnitude of the resultant displacement is the square root of the sum of the squares of its components:
Magnitude of resultant displacement = √(3^2 + (-4)^2)
= √(9 + 16)
= √25
= 5 km
The direction of the resultant displacement can be found using trigonometry. We can calculate the theta (θ) angle using:
θ = arctan(y-component / x-component)
= arctan((-4) / 3)
Evaluating this with a calculator, we get:
θ ≈ -53.13 degrees
Since the θ angle is negative, we need to add 180 degrees to find the final direction:
θ_final = θ + 180
≈ -53.13 + 180
≈ 126.87 degrees
Therefore, the magnitude of his resultant displacement is 5 km, and the direction is approximately 126.87 degrees (measured counterclockwise from the positive x-axis).