Which of the following could not be the resultant displacement of adding a 20 metre vector and a 30 metre vector?

Question

BlackTom AI · Accepted Answer

Vector magnitudes obey the triangle inequality: |A − B| ≤ R ≤ A + B. For two vectors of lengths 20 m and 30 m this gives 30 − 20 = 10 ≤ R ≤ 20 + 30 = 50. Therefore any resultant magnitude must lie between 10 m and 50 m (inclusive).

Step-by-step reasoning and calculations:
1. Using extremes:
   - Maximum resultant occurs when the two vectors are parallel and in the same direction: Rmax = 20 + 30 = 50 m.
   - Minimum resultant occurs when they are collinear but opposite: Rmin = |30 − 20| = 10 m.
   Thus 0 m is outside this range and cannot occur.

2. Verify possibility of each option using the vector law of cosines (for completeness):
   - General formula: R^2 = 20^2 + 30^2 + 2·20·30·cosθ = 400 + 900 + 1200·cosθ = 1300 + 1200·cosθ.
   - For R = 25 m: 25^2 = 625. Solve for cosθ: cosθ = (625 − 1300) / 1200 = −675 / 1200 = −0.5625, which is between −1 and 1, so an angle exists and 25 m is achievable.
   - For R = 50 m: 50^2 = 2500 → cosθ = (2500 − 1300) / 1200 = 1200 / 1200 = 1, corresponding to θ = 0° (vectors aligned), so 50 m is achievable.
   - For R = 10 m: 10^2 = 100 → cosθ = (100 − 1300) / 1200 = −1200 / 1200 = −1, corresponding to θ = 180° (opposite directions), so 10 m is achievable.
   - For R = 0 m: 0^2 = 0 → cosθ = (0 − 1300) / 1200 = −1.0833..., which is impossible because cosθ must be between −1 and 1.

Why the correct answer is correct: 0 m lies below the minimum possible resultant magnitude (10 m), so it cannot be the resultant of a 20 m and a 30 m vector.

Therefore the impossible resultant is option a. 0 metre. This confirms the provided answer.

Which of the following could not be the resultant displacement of adding a 20 metre vector and a 30 metre vector?单项选择题

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