Solution:

Vector Algebra Method:

In this technique, each force is broken down into its horizontal and vertical components, which we label as Fx,i and Fy,i for the itch force.

The resultant’s horizontal portion Rx is the sum of the horizontal components from all the individual forces that are present:

$$ R_x=\sum_{i=1}^N F_{x, i} $$

Likewise, we separately sum the vertical components by using the equation,

$$ R_y=\sum_{i=1}^N F_{y, i} $$

The resultant force is then expressed in vector form as R = Rxi + Ryj. Similar to Equation, we apply the expressions to calculate the resultant’s magnitude R and direction .

$$ \begin{aligned} R &=\sqrt{R_x^2+R_y^2} \\ \theta &=\tan ^{-1}\left(\frac{R_y}{R_x}\right) \end{aligned} $$

As before, the actual value is found after considering the positive and negative signs of Rx and Ry, so that lies in the correct quadrant.

Vector Polygon Method:

An alternative technique for finding the cumulative influence of several forces is the vector polygon method.

The resultant of a force system can be found by sketching a polygon to represent the addition of the Fi vectors.

The magnitude and direction of the results are determined by applying rules of trigonometry to the polygon’s geometry.

Referring to the mounting post in Figure, the vector polygon for those three forces is drawn by adding the individual Fi’s in a chain according to the head-to-tail rule.

The order in which the forces are drawn on the diagram does not matter insofar as the final result is concerned, but diagrams will appear visually different for various addition sequences.

The endpoint is located at the tip of the last vector added to the chain.

As indicated in Figure (b), the resultant R extends from the start of the chain to its end. The action of R on the bracket is entirely equivalent to the combined effect of the three forces acting together.

Finally, the magnitude and direction of the results are determined by applying trigonometric identities to the polygon’s shape.

Some of the relevant equations for right and oblique triangles are reviewed in Appendix B.