Resultant of forces meeting at a point — compute magnitude from components If several forces act at a point, the magnitude of the single resultant R in terms of the algebraic sums of horizontal and vertical components (ΣH and ΣV) is:

Introduction / Context:
Vector addition of forces is fundamental. When multiple forces act at a point, we commonly resolve them into orthogonal components and then recombine to find a single equivalent resultant force.

Given Data / Assumptions:

• All forces are concurrent (meet at a single point).
• Components are resolved along mutually perpendicular axes (horizontal H and vertical V).
• ΣH and ΣV denote algebraic sums of components.

Concept / Approach:
The resultant of orthogonal components follows the Pythagorean relationship. If R has components (ΣH, ΣV), then the magnitude of R is given by R = sqrt( (ΣH)^2 + (ΣV)^2 ). The direction θ satisfies tan(θ) = ΣV / ΣH.

Step-by-Step Solution:

Verification / Alternative check:
Check special case: If ΣH = 0, then R = |ΣV|; if ΣV = 0, then R = |ΣH|. This is consistent with the square root expression.

Why Other Options Are Wrong:

• (ΣH)^2 + (ΣV)^2 without a square root: Gives the squared magnitude, not the magnitude.
• (ΣH + ΣV)^2 or ΣH + ΣV: Mixing components algebraically destroys vector nature.
• √(ΣH + ΣV): Nondimensional and incorrect.

Common Pitfalls:
Forgetting to take the square root; sign errors when summing components; incorrect angle quadrant due to signs of ΣH and ΣV.

Final Answer:
R = √( (ΣH)^2 + (ΣV)^2 )