There are many methods for adding vectors and which method you use depends on the information you know about the vectors. The resultant is the result from adding vectors. The three main methods are[br][list][*]Parallelogram method: The easiest method if you are adding vectors graphically.[/*][*]Triangle rule: Great if you know vector magnitudes and angles and need resultant magnitude and angle.[/*][*]Cartesian components: Most general method, handy for adding more than two vectors. Also, works well in three-dimensions.[/*][/list]
This rule only involves drawing parallel lines. Below are two vectors you can change by moving the points A and B. You can then step through the process with the single play buttons with the steps described below.
[list=1][*]Move the points A and B to change the vectors to be added.[/*][*]Draw a line parallel to vector V[sub]A[/sub][/*][*]Draw a line parallel to vector VB[/*][*]Note the intersection point of the two parallel lines. (HaHa)[/*][*]Draw the resultant vector from the origin to the intersection point.[/*][/list]To add more vectors just add them two at a time. The vectors should originate at the same point.
This method is handy if you are working with magnitudes and angles of vectors. This is sometimes called the tip-to-tail method. The steps are outlined below.
[list=1][*]Here you can set the vectors you want to add by moving the points. Note the identification of the angles. Different angles may be known and that would change the procedure slightly.[br]Here use the slider to move the tail of vector V[sub]B[/sub] to the tip of vector V[sub]A[/sub]. Leave the slider at 1 to continue.[/*][*]The resultant then goes from the tail of vector V[sub]A[/sub] to the tip of vector V[sub]B[/sub].[/*][*]Note the angle [math]\alpha[/math] is an opposite inside angle for a line parallel to the x-axis.[/*][*]Note the angle [math]\beta[/math] is a corresponding angle for the line parallel to the x-axis. The inside angle of the triangle of vectors is then just [math]\alpha[/math]+[math]\beta[/math]. The magnitude of the resultant can then be calculated with the Law of Cosines, [math]Resultant = \sqrt{V_A^2 + V_B^2 - 2 V_A V_B \cos(\alpha+\beta)}[/math][/*][*]The triangle inside angle at the origin can then be calculated from the Law of Sines, [math]\gamma=\sin^{-1}\left(\frac{V_B}{Resultant}\sin\left(\alpha+\beta\right)\right)[/math]. Thus the magnitude and angle of the sum of vectors is [math]Resultant\angle\left(\alpha+\gamma\right)[/math]. The actual angle relations may change depending on the vectors and angles known at the beginning.[/*][/list]
Before going into the component method lets go over how to calculate the components of a vector. The method to use depends on what information is known about the vector. Below is a vector from point A to point B called [i]r[sub]AB[/sub][/i]. You can freely move the points. The components in the x and y directions are also shown as [i]ABx[/i] and [i]ABy[/i] respectively. Above that is how to calculate the components based knowing on the angle [math]\theta[/math] indicated. These could also be used to calculate the vector magnitude if you knew a component and angle. [br][size=150][color=#ff0000]Warning: Here Cartesian reference sign convention is being used. Other sign conventions may be used in diagrams.[/color][/size][br][br]Move the points A and B and note how the formulas change depending on the angle [math]\theta[/math]. This is really on applying the definitions of sine and cosine. The nemonics SOH CAH TOA (Sine is Opposite over Hypotneuse, Cosine is Adjacent over Hypoteneuse, Tangent is Opposite over Adjacent) or Cosine is Cozy can help in remembering whether to use sin or cosine. Make sure to move B all the way around A.[br][br]
If you do not know the angle but you know the direction of the vector goes from a point A to a point C the components can be calculated more easily using a unit vector approach.
In this method you know a horizontal (x axis) distance and a vertical (y axis) distance between points A and C. These can be used as component values of the vector from A to C, noting the direction as positive or negative. Then calculating the distance from A to C using the Pythagorean Theorem the ratios of the triangle sides remain constant as as the triangle size changes. Thus the components of a vector in the same direction can be determined from the ratios.[br][math]\frac{AB_x}{\left|r_{AB}\right|}=\frac{AC_x}{L_{AC}}\text{ and }\frac{AB_y}{\left|r_{AB}\right|}=\frac{AC_y}{L_{AC}}[/math] . Note the signs of the components of r[sub]AB[/sub] would be the same as the components of the vector from A to C.[br]It is also possible to use this method if the any two of the distances for the triangle formed from the Vector A to C and its components.[br][br]The slider, a, can be used to change the magnitude of r[sub]AB[/sub] .
Once vectors are decomposed into components the components can be added separately to determine the components of the resultant. This is very convenient when adding more than two vectors or when working in three dimensions.
The vectors V[sub]A[/sub] and V[sub]B[/sub] are shown as well as their components along each axis. [br]Move the points to change the vectors V[sub]A[/sub] and V[sub]B[/sub].[br]Move the slider Tip2Tail all the way to 1 to connect the vectors tip to tail. This will show the resultant vector also.[br]Move the Connect slider to 1 to move the y components to connect to the tip of the x components.