11 Summed Formulas

The following formulas are the summed variations of the previously discussed equations. Parallel to chapter five, these are the equations that should be found on your formula sheet.

The first formula is the basic summation for the resultant in the [latex]x,\ y[/latex], and [latex]z[/latex] plane. This is the formula that you have been using this entire time, put onto paper:

[latex]{\vec{R}}_{xyz}=\sum_{i=1}^{n}\sqrt{\left(f_{iz}\right)^2+\left(f_{ix}\right)^2+\left(f_{iz}\right)^2}[/latex]

This, and the following two formulas are my personal choice for your formula sheet. These are super compact, and if you can remember what the values within the brackets stand for then these are the formulas for you.

This time it is a bit different. We have two formulas for this system. Remember how we had to find two resultant angles? One in the [latex]x,\ y,[/latex] and one in the [latex]x,\ y,[/latex] and [latex]z[/latex]. It is not quite a possibility to run one equation and receive two values, so I have come up with the following:

[latex]\theta_{Rxy}=\sum_{i=1}^{n}{{tan}^{-1}\left(\frac{f_{iy}}{f_{ix}}\right)}[/latex]

and,

[latex]\theta_{\vec{R}xyz}=\sum_{i=1}^{n}{{tan}^{-1}\left(\frac{f_{zi}}{\sqrt{\left(f_{xi}\right)^2+\left(f_{yi}\right)^2}}\right)}[/latex]

Once more these are my selections for what ends up on your formula sheet. However, if you prefer something that just tells you what is inside the brackets, then you may be fonder of these:

[latex]{\vec{R}}_{xyz}=\sum_{i=1}^{n}\sqrt{\left(\vec{f_i}sin\theta_{ixyz}\right)^2+\left(\left(\vec{f_i}cos\theta_{ixyz}\right)cos\theta_{ixy}\right)^2+\left(\left(\vec{f_i}cos\theta_{ixyz}\right)sin\theta_{ixy}\right)^2}[/latex]

[latex]\theta_{Rxy}=\sum_{i=1}^{n}{{tan}^{-1}\left(\frac{\left(\vec{f_i}cos\theta_{ixyz}\right)sin\theta_{ixy}}{\left(\vec{f_i}cos\theta_{xyzi}\right)cos\theta_{xyi}}\right)}[/latex]

[latex]\theta_{\vec{R}xyz}=\sum_{i=1}^{n}{{tan}^{-1}\left(\frac{\vec{f_i}sin\theta_{xyzi}}{\sqrt{\left(\left(\vec{f_i}cos\theta_{xyzi}\right)cos\theta_{xyi}\right)^2+\left(\left(\vec{f_i}cos\theta_{xyzi}\right)sin\theta_{xyi}\right)^2}}\right)}[/latex]

These formulas show you what exactly is within each and every bracket, in case you forget. Personally, I find that this is the best use for test formula sheets, as sometimes the pressure or stress can make you forget, and this could act as your saving grace in your time of need.

Furthermore, as per usual, I will provide the complete and expanded equations to show my thought process in its entirety:

[latex]{\vec{R}}_{xyz}=\sum_{i=1}^{n}\sqrt{\left(sin\theta_{ixyz}=\frac{f_{iz}}{\vec{f_i}}\right)^2+\left(cos\theta_{ixy}=\frac{f_{ix}}{cos\theta_{ixyz}=\frac{f_{ixy}}{\vec{f_i}}}\right)^2+\left(sin\theta_{ixy}=\frac{f_{iy}}{cos\theta_{ixyz}=\frac{f_{ixy}}{\vec{f_i}}}\right)^2}[/latex]

[latex]\theta_{Rxy}=\sum_{i=1}^{n}{{tan}^{-1}\left(\frac{sin\theta_{ixy}=\frac{f_{iy}}{cos\theta_{ixyz}=\frac{f_{ixy}}{\vec{f_i}}}}{cos\theta_{ixy}=\frac{f_{ix}}{cos\theta_{ixyz}=\frac{f_{ixy}}{\vec{f_i}}}}\right)}[/latex]

[latex]\theta_{\vec{R}xyz}=\sum_{i=1}^{n}{{tan}^{-1}\left(\frac{sin\theta_{ixyz}=\frac{f_{iz}}{\vec{f_i}}}{\sqrt{\left(cos\theta_{ixy}=\frac{f_{ix}}{cos\theta_{ixyz}=\frac{f_{ixy}}{\vec{f_i}}}\right)^2+\left(sin\theta_{ixy}=\frac{f_{iy}}{cos\theta_{ixyz}=\frac{f_{ixy}}{\vec{f_i}}}\right)^2}}\right)}[/latex]

Now, they may look intimidating, but do not fret. Though these puppies are big, if you take a moment and look, you can see what is going on inside. These equations contain nothing but everything that we have done in previous chapters to get our answer.