A boy walking to school walks 3 km north and then 4 km west. His displacement is

Question

BlackTom AI · Accepted Answer

Pythagorean theorem applies because the 3 km north and 4 km west legs are perpendicular, so the displacement magnitude is the hypotenuse of a right triangle.

1. Relevant knowledge points and concepts
- Displacement is a vector: it has magnitude and direction.
- For perpendicular vector components, magnitude = sqrt(x^2 + y^2) (Pythagorean theorem).
- Direction can be found with trigonometry: tan(theta) = opposite/adjacent. Bearing NθW means angle θ west of north.

2. Step-by-step reasoning
- Take north as +y and east as +x. The boy goes +3 km in y and -4 km in x, so the resultant vector is (-4, +3).
- Magnitude = sqrt((-4)^2 + (3)^2).
- Angle relative to north: use tan(theta) = |x|/|y| because we measure angle west of north.

3. Calculations (all intermediate steps)
- Compute squares: (-4)^2 = 16, (3)^2 = 9.
- Sum: 16 + 9 = 25.
- Magnitude: sqrt(25) = 5 km.
- Angle: theta = arctan(|x|/|y|) = arctan(4/3) ≈ 53.13°, which rounds to 53°.
- Therefore the direction is N53°W (north then 53° toward the west).

4. Why the correct answer is correct
- The magnitude 5 km follows directly from the 3-4-5 right triangle. The orientation (-4 east, +3 north) corresponds to a bearing of north 53° west, so "5 km N53°W" fully specifies the displacement vector.

5. Why the other options are incorrect
- a. 1 km: incorrect — not the vector difference or magnitude (3 and 4 do not subtract to 1 for displacement magnitude).
- b. 5 km: gives correct magnitude but omits direction; the question asks for displacement (a vector), so direction matters.
- d. 7 km: 7 km is the total path length (3 + 4) but not the straight-line displacement.

6. Conclusion
The displacement is 5 km toward N53°W, which confirms the provided correct answer: c. 5 km N53°W.

A boy walking to school walks 3 km north and then 4 km west. His displacement is单项选择题

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