Vector Triangle Inequality

Steps:

• For three forces to sum to zero, their magnitudes must satisfy the triangle inequality: each ≤ sum of the other two.
• Check: 2 + 4 = 6 < 7, so the 7 N force exceeds the sum of the others.
• Minimum resultant magnitude is 7 - (2 + 4) = 1 N; maximum is 2 + 4 + 7 = 13 N.
• Thus, possible resultants range from 1 N to 13 N.

Why A is correct:

• Zero resultant requires forces to form a closed triangle, but 7 > 2 + 4 violates the triangle inequality theorem for vectors.

Why the others are wrong:

• B: 5 N falls within the 1–13 N range, achievable by adjusting directions.
• C: 8 N falls within the 1–13 N range, achievable by adjusting directions.
• D: 13 N is the maximum, when all forces align in the same direction.

Final answer: A