Resultant of Two Forces – Magnitude Formula\nTwo forces P and Q act at an angle θ to each other. What is the magnitude of their resultant R?

Introduction / Context:
The parallelogram (or triangle) law of forces gives the resultant magnitude and direction when two vectors act at an angle θ. This is foundational in statics and dynamics.

Given Data / Assumptions:

• Two vectors (forces) of magnitudes P and Q.
• Included angle between them is θ.

Concept / Approach:
Resolve vectors or use the cosine rule on the triangle formed by P, Q, and the resultant R. The cosine rule for sides a, b with included angle C gives c^2 = a^2 + b^2 − 2ab cos C; adapting signs based on geometry yields the standard resultant formula.

Step-by-Step Solution:
Construct the parallelogram of forces with sides P and Q. Diagonal from the common point represents R. By geometry: R^2 = P^2 + Q^2 + 2 P Q cos θ (when θ is the included angle between P and Q tail-to-tail). Thus R = sqrt(P^2 + Q^2 + 2 P Q cos θ).

Verification / Alternative check:
Component method: Rx = P + Q cos θ, Ry = Q sin θ ⇒ R = sqrt((P + Q cos θ)^2 + (Q sin θ)^2) which simplifies to the same expression.

Why Other Options Are Wrong:
Linear sum P + Q + θ has wrong dimensions. The minus-sign cosine form corresponds to the side opposite the included angle in a different arrangement (effectively P and −Q). Other algebraic forms do not follow from vector addition.

Common Pitfalls:
Confusing included angle and supplementary angle; always base on the geometry used.

Final Answer:
R = sqrt(P^2 + Q^2 + 2 P Q cos θ)