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Difficult to draw
Of course, we can't easily draw that vector diagram on a flat sheet of paper or construct it on a flat graph board, but we can solve for the displacementvecAD if we are familiar with computations in three dimensions. (An explanation of 3D computations is beyond the scope of this module.)
The parallelogram law
There is a law that states:
The sum of two vectors in a plane is represented by the diagonal of a parallelogram whose adjacent sides represent the two vector quantities.
The resultant
As I mentioned earlier, the sum of two or more vectors is called their resultant .
In general, the resultant is not simply the algebraic or arithmetic sum of the vector magnitudes. Instead, in our original vector equation
vecAC = vecAB + vecBC
vecAC is the resultant of the sum of vecAB and vecBC in the sense that a single vector vecAC would have the same effect as vecAB and vecAD actingjointly.
Geometrical addition
The parallelogram law is a form of geometrical addition. In physics, it is often used to find graphical solutions to problems involving forces, velocities,displacements, accelerations, electric fields, and other directed quantities. (Directed quantities have both magnitude and direction.)
Solving a vector problem with a parallelogram
Pretend that you walk from point A
Let's use your graph board and the parallelogram law to find the resultant displacement vector vecAC.
Tactile graphics
The svg file for this exercise is named Phy1060e1.svg. The table of key-value pairs for this file is provided in Figure 9 .
| Figure 9 . Key-value pairs for the image in Phy1060e1.svg. |
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m: Adding vectors using the parallelogram law.
n: Bo: C
p: Aq: Vector B C
r: Resultant vectors: Vector A B
t: Line parallel to vector A Bu: Line parallel to vector B C
v: File: Phy1060e1.svg |
The image contained in this file is shown in Figure 10 . A non-mirror-image version of the file is shown in Figure 18 .
| Figure 10 . Mirror image contained in the file named Phy1060e1.svg. |
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Put both vector tails at point A
Mentally designate the starting point on your graph board as point A. Construct a vector with a length of 5 meters and an angle of 0 degrees relativeto the horizontal with its tail at point A. Mentally designate this vector as vecAB.
Construct a second vector with a length of 6 meters at an angle of 30 degrees with its tail at point A. Mentally designate this vector as vecBC.
Parallel lines
Now comes the tricky part. Using a rubber band and a push pin, construct a line at least 6 meters long starting at the head of vecBC. Make this lineparallel to vecAB.
(There are drawing tools that make it easy for sighted students to draw a linethat is parallel to another line. This is one reason why this is a popular technique for graphically adding vectors.)
Construct another line, at least 7 meters long, starting at the head of vecAB. Make this line parallel to vecBC.
A parallelogram
Push a pin into the graph board at the intersection of the two lines.
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