Statics – Resolution of Forces and Resultant In engineering mechanics, the resolved component of the resultant of two forces (inclined at an angle θ to each other) taken along a specified direction is equal to which of the following?

Introduction / Context:
This question tests the fundamental resolution principle in statics. When combining multiple forces, any single component of the resultant along a chosen axis equals the algebraic sum of the corresponding components of the individual forces along that same axis. This principle underpins free-body diagram analysis, equilibrium checks, and vector addition in mechanics.

Given Data / Assumptions:

• Two forces act at an angle θ to one another.
• A specific reference direction (axis) is chosen.
• Standard right-hand sign convention for components.

Concept / Approach:

The vector resultant R of forces F1 and F2 is obtained by vector addition. Components are linear operations on vectors. Therefore, the component of R along any direction is simply the component-wise sum: R_dir = F1_dir + F2_dir. This holds regardless of the angle θ between forces or their magnitudes, provided signs (algebraic sense) are respected.

Step-by-Step Solution:

Verification / Alternative check:

Construct a vector diagram or use Cartesian components: R = (F1x + F2x) i + (F1y + F2y) j. Projection of R on any axis equals the sum of projections of F1 and F2 on that axis.

Why Other Options Are Wrong:

(b) Ignores direction and signs; not generally valid. (c) Applies only to a special projection relationship and even then not universally. (d) and (e) combine magnitudes with trigonometric factors incorrectly; components must be taken force-by-force before summing.

Common Pitfalls:

Mixing scalar magnitudes with vector components; forgetting to use algebraic signs for directions; using a single common angle θ when individual force angles to the reference differ.

Final Answer:

Algebraic sum of the resolved components of the two forces along that direction