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

Triangle inequality for vectors gives the allowed range for the magnitude of a sum: if two vectors have magnitudes A and B, then |A - B| ≤ |A + B| ≤ A + B. Here A = 20 m and B = 30 m, so any resultant magnitude R must satisfy:

1) Relevant concepts
- Triangle inequality and the law of cosines for vector addition.
- Magnitude bounds: minimum is the absolute difference, maximum is the sum.
- For a given R, the angle between vectors θ can be found from the law of cosines: R^2 = A^2 + B^2 + 2AB cosθ, which requires −1 ≤ cosθ ≤ 1.

2) Step-by-step reasoning and intermediate calculations
- Apply the bounds: |30 − 20| ≤ R ≤ 30 + 20 → 10 ≤ R ≤ 50. So any possible resultant magnitude must lie between 10 m and 50 m inclusive.
- Test each option:
  a) R = 0 m: 0 is outside the allowed range (0 < 10), so impossible.
  b) R = 50 m: check cosθ from law of cosines:
     cosθ = (R^2 − A^2 − B^2) / (2AB) = (2500 − 400 − 900) / (1200) = 1200/1200 = 1 → θ = 0°, vectors aligned. Valid.
  c) R = 25 m: cosθ = (625 − 400 − 900) / 1200 = −675/1200 = −0.5625 → θ ≈ 124.2°. This is within [0°,180°], so valid.
  d) R = 10 m: cosθ = (100 − 400 − 900) / 1200 = −1200/1200 = −1 → θ = 180°, vectors opposite. Valid.

3) Why the correct answer is correct
- Option (a) 0 m violates the triangle inequality because the smaller vector (20 m) cannot completely cancel the larger (30 m); the smallest possible resultant is 10 m (30 − 20). The law of cosines gives cosθ = −1.0833 for R = 0, which is impossible since cosθ ∈ [−1,1].

4) Why the other options are not the correct choice here
- (b) 50 m is achievable when vectors point the same way.
- (c) 25 m is achievable for an obtuse angle (~124.2°) between vectors.
- (d) 10 m is achievable when vectors are exactly opposite.

Conclusion: Only 0 metre is impossible, so the correct answer is (a) 0 metre, matching the provided answer.

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

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