Direction of the resultant of two forces Two coplanar forces P and Q act at an angle θ to each other. If the resultant makes an angle α with the direction of P, what is tan α in terms of P, Q, and θ?

Introduction / Context:
When combining two forces by the parallelogram rule, the direction of the resultant relative to one of the forces is obtained by resolving components along and perpendicular to that force. This is a standard vector-resolution result.

Given Data / Assumptions:

• Two forces: P (reference direction) and Q at angle θ to P.
• Resultant makes angle α with P.
• Coplanar, concurrent forces.

Concept / Approach:
Resolve Q into components parallel and perpendicular to P. The resultant R has components R_parallel = P + Q cos θ and R_perp = Q sin θ. Therefore tan α is the ratio of the perpendicular to parallel components.

Step-by-Step Solution:

Verification / Alternative check:
Construct parallelogram and measure α; the component formula matches the geometry. Dimensionless ratio is consistent.

Why Other Options Are Wrong:

• Swaps P and Q or swaps sine and cosine incorrectly.
• Reciprocal form gives cot α, not tan α.
• Some include wrong signs or denominators.

Common Pitfalls:
Using θ from the wrong side; forgetting which component is parallel/perpendicular to P.

Final Answer:
tan α = (Q sin θ) / (P + Q cos θ)