Definition check: The shear force at a point on a beam equals the algebraic __________ of all the forces on either side of that point.

Introduction / Context:
Shear force (V) and bending moment (M) calculations start from statics. The basic definition of shear at a section is vital for drawing shear force diagrams correctly and for understanding internal force resultants in beams.

Given Data / Assumptions:

• Beam in equilibrium subjected to external loads and reactions.
• Section cut at a point x along the span.
• Sign convention for shear is consistent along the analysis.

Concept / Approach:
The shear force at a section equals the algebraic sum of all external forces (loads and reactions) acting on one side of the section. Choosing the left or right side consistently yields the same magnitude with appropriate sign, assuming equilibrium.

Step-by-Step Solution:

Verification / Alternative check:
Equilibrium requires ΣF_y = 0 for the whole beam; hence shear computed from either side must match. This consistency check helps catch arithmetic or sign errors.

Why Other Options Are Wrong:
Difference, product, ratio, or average do not represent the equilibrium definition that underpins shear at a section.

Common Pitfalls:
Mixing sign conventions; forgetting to include distributed load resultants over a cut length; ignoring applied moments which affect bending moment but not shear directly.

Final Answer:
sum