Step 1: Conceptual Foundation:
Equilibrium for a point body (or a point on a rigid body) subjected to forces necessitates that the net force, which is the vector sum of all applied forces, must be zero. This principle aligns with Newton's First Law. The objective is to determine the conditions under which two forces, \( \vec{F}_1 \) and \( \vec{F}_2 \), result in a zero vector sum.
Step 2: Governing Equation:
The condition for equilibrium is mathematically expressed as:
\[ \vec{F}_{net} = \vec{F}_1 + \vec{F}_2 = \vec{0} \]
This vector equation dictates specific requirements for the magnitudes and directions of the two forces.
Step 3: Analytical Breakdown:
From the equilibrium equation \( \vec{F}_1 + \vec{F}_2 = \vec{0} \), it follows that \( \vec{F}_1 = -\vec{F}_2 \). This relationship can be dissected as follows:
Magnitude Requirement: The magnitude of a vector is a non-negative scalar value. Applying the magnitude operation to both sides of the equation yields \( |\vec{F}_1| = |-\vec{F}_2| = |\vec{F}_2| \). Consequently, the magnitudes of the two forces must be equal. Therefore, statement (A) is validated as true, while statement (E) is deemed false.
Directional Requirement: The negative sign in \( \vec{F}_1 = -\vec{F}_2 \) signifies that vector \( \vec{F}_1 \) possesses the exact opposite direction to vector \( \vec{F}_2 \). This implies that the forces must act in opposing directions. Accordingly, statement (C) is confirmed as true, and statement (B) is identified as false.
Collinearity Requirement: For two vectors to be precisely opposite, they must be aligned along the same straight line. This property is referred to as collinearity. Hence, statement (D) is established as true.
Step 4: Conclusion:
For two forces to achieve equilibrium, they must satisfy three criteria: they must be equal in magnitude, opposite in direction, and act along the same straight line (collinear). Consequently, statements (A), (C), and (D) are all indispensable for equilibrium.