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Chapter 3: Problem 24

Margaret walks to the store using the following path: 0.500 miles west, 0.200 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? Use vector components to find the answer.

Short Answer

Expert verified
Answer: Margaret's total displacement is approximately 0.2828 miles in the southwest direction at 45° from the west direction.

Step by step solution

01

Identify the vector components for each direction

Since Margaret walks west, north, and east, her path can be represented by three vectors: - Vector A: 0.500 miles west (negative x-direction) - Vector B: 0.200 miles north (positive y-direction) - Vector C: 0.300 miles east (positive x-direction)
02

Calculate the total displacement vector

Now we can find the total displacement vector by adding the vector components: Vector A: (-0.500, 0) Vector B: (0, 0.200) Vector C: (0.300, 0) Total Displacement Vector (D) = Vector A + Vector B + Vector C = (-0.500 + 0.300, 0 + 0.200) = (-0.200, 0.200)
03

Calculate the magnitude and direction of the displacement vector

To find the magnitude (length) of the displacement vector, we can use the Pythagorean theorem: Magnitude of D = sqrt((-0.200)^2 + (0.200)^2) = sqrt(0.04 + 0.04) = sqrt(0.08) = 0.2828 miles (approx.) To find the direction of the displacement vector, we can use the arctangent function (angle of the vector with the positive x-axis): Direction of D = arctan(0.200/-0.200) = arctan(-1) = -45° Since the angle is negative, this indicates that the direction is 45° below the negative x-axis (west). Thus, the direction of the displacement vector is 45° southwest.
04

Present the final answer

Margaret's total displacement is approximately 0.2828 miles in the southwest direction at 45° from the west direction.

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