Final answer:
The third force needed to balance the first two forces in a Physics problem must have a magnitude and direction opposite to the vector sum of the two known forces. To find it, you negate the sum of the forces' components, with specifics depending on the exact magnitudes and directions of the original forces.
Step-by-step explanation:
The student is asking how to calculate the third force needed to balance two other forces acting on an object, which is a problem in Physics involving vector addition.
The forces mentioned seem to be two-dimensional, having both i and j components, which means they act in the horizontal (x) and vertical (y) directions on a flat plane.
To achieve a balanced state, where the object is not accelerating, the sum of all forces acting on it must be zero; this is known as a condition of equilibrium. Here, the third force must be equal and opposite to the vector sum of the first two forces.
Assuming the known forces are F₁ and F₂, the third force, F₃, required to balance them would be calculated as F₃ = -(F₁ + F₂).
The exact values of F₁ and F₂ are not provided in the question, so specific numerical answers cannot be given, but the provided choices imply that the result should have both components negated and involve the square root of 2, likely because the resultant force must be along a diagonal in the opposite direction with specific magnitudes.