A person walks 454 m East, followed by 337.9 m South and finally 1446.9 m West. What is this person's displacement vector taking reference from the starting position?

Question

BlackTom AI · Accepted Answer

To begin, I will track the net displacement by treating east as +x and north as +y, then combine the given legs.
First leg: 454 m East adds +454 to x, so x = +454. The y component remains 0.
Second leg: 337.9 m South subtracts from y, so y = -337.9, while x stays +454.
Third leg: 1446.9 m West subtracts from x, so x = 454 - 1446.9 = -992.9. The y component is still -337.9.
Thus, the final displacement vector from the start is Δx = -992.9 m and Δy = -337.9 m, placing the point in the third quadrant (west and south of the origin).
Now compute the magnitude of this displacement using the Pythagorean theorem: |Δr| = sqrt(Δx^2 + Δy^2) = sqrt((-992.9)^2 + (-337.9)^2).
Carrying out the squares: (-992.9)^2 ≈ 985,850.41 and (-337.9)^2 ≈ 114,176.41. Adding gives ≈ 1,100,026.82. Taking the square root yields |Δr| ≈ 1,049 m (rounded to the nearest meter).
Next, determine the direction. The vector points primarily west with a south component. The angle south of west is arctan(|Δy|/|Δx|) = arctan(337.9/992.9) ≈ arctan(0.340) ≈ 18.8 degrees.
Therefore, the displacement is approximately 1,049 m in a direction about 18.8° south of west. An alternative way to express the bearing is from East, clockwise, about 198.8° (since 180° is west and you go an additional 18.8° toward the south).
Note on rounding: if you use slightly different intermediate rounding, you may see a value near 1,048–1,049 m and ~18.8°; both convey the same direction and magnitude.

MUF0121 Physics Unit 1 - Semester 2, 2025

A person walks 454 m East, followed by 337.9 m South and finally 1446.9 m West. What is this person's displacement vector taking reference from the starting position?

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