To determine the resultant of multiple forces, you must combine them as vectors, carefully accounting for both their magnitudes and directions.

In physics, forces are classified as vector quantities, which means they possess both magnitude (size) and direction. When multiple forces act on an object, it is insufficient to merely sum their magnitudes to ascertain the total force. Instead, you must treat these forces as vectors, incorporating both their magnitudes and directions. The total force resulting from this vector addition is referred to as the resultant force.

To add forces as vectors, two common methods can be employed: the parallelogram method and the triangle method.

1. Parallelogram Method: This involves constructing a parallelogram where the forces are represented as adjacent sides. The resultant force is depicted by the diagonal of the parallelogram.

2. Triangle Method: In this approach, you create a triangle using the forces as sides. The resultant force is represented by the third side of the triangle, drawn from the tail of the first vector to the head of the second.

When the forces are aligned along the same line, you can simply add or subtract their magnitudes based on their directions. However, if the forces act at angles to one another, trigonometric methods are necessary to find the resultant force. You can decompose each force into its components along the $x$ and $y$ axes, sum these components separately, and then apply Pythagoras’ theorem to compute the magnitude of the resultant force. The direction of the resultant can be determined using trigonometric functions.

In certain situations, especially when forces change over time or act in three dimensions, more advanced mathematical techniques, including calculus, may be required to find the resultant force. Nevertheless, the fundamental principle remains consistent: add the forces as vectors, taking into account both their magnitudes and directions.