Margaret walks to the store using the following path: 0.420 miles west, 0.265 miles north, 0.300 miles east. What is her total displacement? That is, what is the length and direction of the vector that points from her house directly to the store?
To find the total displacement, we need to calculate the net displacement by adding all the individual displacements together.
First, let's assign positive and negative values to the directions. Let's consider east as positive (to the right) and west as negative (to the left), and north as positive (up).
The first displacement is 0.420 miles west, so it has a value of -0.420 miles in the x-axis (horizontal direction).
The second displacement is 0.265 miles north, so it has a value of +0.265 miles in the y-axis (vertical direction).
The third displacement is 0.300 miles east, so it has a value of +0.300 miles in the x-axis.
To find the net displacement, we add the x and y components of the displacements:
Net displacement in x-axis = -0.420 + 0.300 = -0.120 miles
Net displacement in y-axis = 0.265 miles
To find the length of the resultant vector, we use the Pythagorean theorem:
Length = √((-0.120)^2 + (0.265)^2) = √(0.0144 + 0.0702) = √0.0846 ≈ 0.291 miles
Now, to find the direction of the vector, we need to calculate the angle it makes with the positive x-axis. We can use trigonometry to find the angle:
θ = arctan(0.265 / -0.120) ≈ -64.48 degrees or 295.52 degrees
Since we're dealing with a vector pointing from her house to the store, we take the angle between her house and the store, which is 295.52 degrees relative to the positive x-axis.
Therefore, her total displacement is approximately 0.291 miles with a direction of approximately 295.52 degrees (counterclockwise from the positive x-axis).