Margaret walks to the store using the path: 0.500 miles West, 0.200 miles North, 0.300 miles East. What is the total displacement (direction and length of the vector that points from her house directly to the store)?
well, net West is .2mi, and N is .2
Looks like length is .2*sqrt2, and the direction is 45deg W of N
12.366
To find the total displacement, we need to determine the direction and length of the vector that points from Margaret's house directly to the store.
First, let's break down the given path into its components:
- Margaret walks 0.500 miles West.
- Margaret walks 0.200 miles North.
- Margaret walks 0.300 miles East.
Now, we can calculate the east-west component and north-south component separately:
- The east-west component is obtained by subtracting the total distance traveled east (0.300 miles) from the total distance traveled west (0.500 miles): 0.500 miles - 0.300 miles = 0.200 miles West.
- The north-south component is obtained by subtracting the total distance traveled south (0 miles) from the total distance traveled north (0.200 miles): 0.200 miles - 0 miles = 0.200 miles North.
Now, we have the components of displacement:
- East-west component: 0.200 miles West.
- North-south component: 0.200 miles North.
To find the total displacement, we can use the Pythagorean theorem:
displacement = √(east-west component^2 + north-south component^2)
displacement = √(0.200 miles^2 + 0.200 miles^2) = √(0.04 square miles + 0.04 square miles) = √(0.08 square miles) ≈ 0.283 miles.
Therefore, the total displacement from Margaret's house directly to the store is approximately 0.283 miles.
To find the total displacement of Margaret from her house to the store, we need to calculate the net vector by adding up the individual vectors.
Given the path she walks:
- 0.500 miles West (negative x-direction)
- 0.200 miles North (positive y-direction)
- 0.300 miles East (positive x-direction)
We can represent the vectors as:
- West vector: -0.500i
- North vector: +0.200j
- East vector: +0.300i
Now, let's add up these vectors to find the net vector:
Net vector = West vector + North vector + East vector
= (-0.500i) + (0.200j) + (0.300i)
To add these vectors, we need to consider their components separately. The final result will have both an x-component (horizontal) and a y-component (vertical).
Net x-component = (-0.500) + (0.300) = -0.200
Net y-component = (0.200)
Therefore, the total displacement (direction and length) that points from Margaret's house directly to the store is a vector with an x-component of -0.200 and a y-component of +0.200. This vector can be represented as:
Total Displacement = (-0.200i + 0.200j)
To find the direction, we can use the trigonometric arctan function, which gives us the angle between the x-axis and the vector.
θ = arctan((Net y-component) / (Net x-component))
θ = arctan(0.200 / -0.200)
θ = arctan(-1)
The trigonometric function arctan(-1) is -45° or -π/4 in radians.
So, the total displacement (direction and length of the vector that points from Margaret's house directly to the store) is 0.282 miles in length and forms an angle of -45° (or -π/4 radians) with the positive x-axis.