Two displacement vectors have magnitudes of 5.0 m and 7.0 m respectively. When these two vectors are added, the magnitude of the resultant

Question

BlackTom AI · Accepted Answer

Vector addition follows the triangle inequality and can be analyzed with the law of cosines: R^2 = A^2 + B^2 + 2AB cosθ, where A and B are magnitudes and θ is the angle between them.

1) Relevant knowledge points and concepts
- Magnitude of resultant R of two vectors of magnitudes A and B satisfies |A − B| ≤ R ≤ A + B (triangle inequality).
- Using the law of cosines gives R^2 = A^2 + B^2 + 2AB cosθ; varying θ from 0 to 180° gives the maximum and minimum R.

2) Step-by-step reasoning
- Let A = 5.0 m and B = 7.0 m.
- Plug into the law of cosines: R^2 = 5^2 + 7^2 + 2(5)(7) cosθ = 25 + 49 + 70 cosθ = 74 + 70 cosθ.
- The cosine term ranges from +1 (vectors parallel, same direction) down to −1 (vectors anti-parallel, opposite directions).

3) Intermediate calculations
- For maximum R (cosθ = +1): R^2 = 74 + 70(1) = 144 → R = sqrt(144) = 12.0 m.
- For minimum R (cosθ = −1): R^2 = 74 + 70(−1) = 4 → R = sqrt(4) = 2.0 m.
- For intermediate angles, R will take values between 2.0 m and 12.0 m.

4) Why the correct answer is correct
- The vector resultant can reach its largest value when the two displacements point the same way (5 + 7 = 12 m) and its smallest when they point opposite each other (|7 − 5| = 2 m). Thus the magnitude could be as small as 2.0 m or as large as 12 m.

5) Why the other options are incorrect
- (a) “is greater than 12 m.” is false because the maximum is 12 m (when vectors align), so it cannot exceed 12 m.
- (b) “is 2.0 m.” is false because 2.0 m is the minimum possible magnitude, not the only value.
- (d) “is 12 m.” is false because 12 m is the maximum possible magnitude, not the only value.

6) Conclusion
- Therefore the correct choice is c. could be as small as 2.0 m or as large as 12 m. This matches the provided answer.

Two displacement vectors have magnitudes of 5.0 m and 7.0 m respectively. When these two vectors are added, the magnitude of the resultantSingle choice

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