Three forces act on the ring. If the resultant force F_R has a magnitude and direction as shown, determine the magnitude and the coordinate direction angles of force F_3.
We will first write each of the forces in Cartesian vector notation. This will make our calculations much easier to perform. Let us first look at the resultant force, F_R We can write the resultant force like so:
We can further simply this.
(To simplify, we expanded the brackets by multiplying each value inside the brackets by 120. Remember the FOIL method)
Now, we will write force F_1 in Cartesian vector notation like this:
Again, we will simplify this like so:
(Again, we expanded the brackets like before)
Next up is force F_2. We can write force F_2 like so:
Finally, we will write the generic Cartesian vector notation for force F_3 as follows:
With that, we can equate the resultant force to the addition of all three other forces.
Now, we will solve for i, j, and the k components.
Remember, we are just solving for each component, same as when we solve for any other equation with x and y variables. In this case, we will equate i groups to the i components, j groups to the j components, and so forth.
Now, we will find the magnitude of force F_3. We can calculate the magnitude using the following equation:
Last step is to find the coordinate direction angles for force F_3.