The resultant of two forces P and Q is least when the angle between them is

Analyze the question: We want the angle θ between forces P and Q that makes their resultant R as small as possible.

Concept: The magnitude of the resultant of two forces is given by the cosine rule (parallelogram law):

R = √(P² + Q² + 2PQ cos θ)

Solution:

R depends on cos θ, so R is smallest when cos θ is smallest.

• θ = 0°: cos θ = 1, so R = P + Q (the maximum)
• θ = 45°: cos θ ≈ 0.707
• θ = 60°: cos θ = 0.5
• θ = 90°: cos θ = 0, so R = √(P² + Q²)
• θ = 180°: cos θ = -1, so R = √(P² − 2PQ + Q²) = |P − Q| (the minimum)

The smallest value of cos θ among the options is -1, at 180°. At this angle the forces act in exactly opposite directions, so they partly cancel and the resultant is just the difference of their magnitudes.

Answer: E. 180°

Exam tip: Remember the two extremes:

• Same direction (0°) gives the maximum resultant, P + Q.
• Opposite directions (180°) gives the minimum resultant, |P − Q|.

Whenever you see "greatest" or "least" resultant, check these two angles first.