5 Summations and Their Formulas

These are the formulas that are the best options to be featured on your formula sheets. Summations, also known as sigma notation, have granted the user the ability to rewrite the equations as many times as needed without filling up their sheet with equations ever increasing in length. My equations use the simplest form of summations allowing jumps in intervals of whole numbers. Though it is possible to write summations increasing in decimal values such as 0.5 or 0.25 by dividing by certain numbers, it is not necessary for my system.

Summations work by telling you your starting point and your endpoint of value [latex]n[/latex]. The subscript is your start, and the superscript is the stop. Therefore, in the following system: [latex]\sum_{i=1}^{4}{3n-2}[/latex] the value of [latex]i[/latex] is dependent on the chosen value within the summed limits. The value of [latex]i[/latex] can be any whole number in between 1 and 4. If you were to change the subscript to a different number, 2 for example, then [latex]i[/latex] would be expressed as any whole number in between 2 and 4. [latex]i[/latex] can be used the same as any other variable; it can be added, subtracted, multiplied, divided. It can even be used as an exponent if needed.

A summation that goes on for an infinite amount of time or as long as the user wants is written with the superscript of [latex]n[/latex]. The value of [latex]n[/latex] represents any real number up to infinity. This is the type of summation we will be using. By using this type of summation, we are able to use a singular equation to solve for multiple vectors, and it just so happens that they are nicely labeled as 1, 2, 3, etc.

Though, for simplicity’s sake, the value of [latex]n[/latex] will not be directly utilized in calculations, it will be used solely for labeling of values. The value of [latex]n[/latex] will correspond to the vector number^[1] and will organize the correct values in the correct positions to allow for a correct answer.

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

Presented above is the reduced summed formula for the vector resultant within my two-dimensional system, and below is the reduced summed formula for the vector resultant angle. You will see shortly why they are known as the reduced variants:

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

The presented formulas are my personal choice for a formula sheet due to their straightforwardness. Written like this the summations tell you exactly what to do and how to do it.

These next formulas technically have different summations due to the way they have been written, but I have included the summation regardless for simplicity’s sake. Please view these next two formulas within the scope of their reduced versions, as it is imperative that the [latex]x[/latex] and [latex]y[/latex] forces are added, not the angles nor the vector forces. The following equations are simply given to show the process in achieving the solutions, and should be read as such:

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

and,

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

Yet again these are the expanded versions of the equation, simply here to demonstrate my thought process and to show my work to its greatest extent. As you can see, all values still maintain their corresponding trig ratios as per their axis. So long as you remember that [latex]x[/latex] is cosine and that [latex]y[/latex] is sine and must be solved accordingly you will have no problems whatsoever.

If you so choose, you can write the formulas as ,

[latex]{\vec{R}}_{xy}=\sum_{i=1}^{n}\sqrt{\left(f_icos\theta_{ixy}\right)^2+\left(f_isin\theta_{ixy}\right)^2}[/latex]

[latex]\theta_{\vec{R}xy}=\sum_{i=1}^{n}{{tan}^{-1}\left(\frac{f_isin\theta_{ixy}}{f_icos\theta_{ixy}}\right)}[/latex]

so long as you maintain the mentality that you are adding the system within the parenthesis, not the individual values.