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parallelogram principle
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(Topic)
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- A starting point for learning vectors is to think that they are directed line segments. Thus a vector
 has a direction and a length (magnitude) and nothing else. Therefore, if two vectors  and  have a same direction and a same length, one can consider them identical (and denote
 ). So the location of a certain vector in the plane (or in the space) is insignificant; in fact one may also think that this vector consists of all possible directed line segments having a common direction and a common length.
- However, a vector
 as an infinite set of directed segments is quite uncomfortable to handle, and one can choose from all possible representants of  one individual directed segment
 , i.e. a line segment directed from a certain point  (the initial point) to another certain point  (the terminal point). Although
 is only a representant of  , one may write
 or
 .
- For describing a vector
 , it's convenient to know its position in the coordinate system of the plane (or the space); there one can say e.g. how great a displacement  means from left to right (i.e. in the direction of  -axis) and how great from below upwards (i.e. in the direction of the  -axis); those displacements may be expressed with two numbers. One may for example
write
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(1) |
where the first (upper) number  tells that the vector leads 5 length-units to the right and the second (lower) number  that it leads 1 length-unit downwards.
Addition of vectors
Since the vector may be interpreted as a combination of a horizontal displacement and a vertical displacement, it's meaningful that by the addition of two vectors the horizontal displacements are summed and likewise the vertical displacements. Accordingly, if we have
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(2) |
then the sum of the vectors (1) and (2) is
which result means a vector leading 6 length-units to the right and 4 down.
When we set the vectors  and  one after the other, as in the above picture, and take the sum vector from the initial point of the first addend to the terminal point of the second addend, then both the horizontal and the vertical displacements are respectively added. The addition rule as a formula using the points is
Note, that the sum vector
 can be also obtained as the diagonal vector of the parallelogram with one pair of opposite sides equal to  and the other pair of opposite sides equal to  . The parallelogram picture illustrates also that the vector addition is commutative, i.e. that
 .
If we think the second (dashed in the third picture) diagonal of the parallelogram, it is halved by the first (blue) diagonal, since the diagonals of any parallelogram bisect each other (see parallelogram theorems); as well the (blue) diagonal representing the sum
 is halved into two equal vectors (better: directed segments)
 . In the triangle  , the vectors  ,  ,  may be called two side vectors and a median vector, all having the common initial point  . Thus we can write the
Theorem. In a triangle, the median vector emanating from a certain vertex is the arithmetic mean of the side vectors emanating from the same vertex.
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"parallelogram principle" is owned by pahio.
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(view preamble)
Cross-references: arithmetic mean, vertex, triangle, parallelogram theorems, diagonal, commutative, parallelogram, sum, addition, numbers, coordinate system, terminal point, initial point, point, line segment, directed segments, infinite set, plane, length, vectors
There are 17 references to this entry.
This is version 13 of parallelogram principle, born on 2008-02-06, modified 2008-08-26.
Object id is 10244, canonical name is ParallelogramPrinciple.
Accessed 2465 times total.
Classification:
| AMS MSC: | 53A45 (Differential geometry :: Classical differential geometry :: Vector and tensor analysis) |
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Pending Errata and Addenda
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 \psline[arrows=->,arro... ...](5.7,-2.2){$\vec{v}$} \rput[a](3.4,-1.8){$\vec{u}\!+\!\vec{v}$} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/10244/l2h/img23.png "\begin{pspicture}(0,0)(7,-4.2) \psdot[linecolor=red](0,0) \psline[arrows=->,arro... ...](5.7,-2.2){$\vec{v}$} \rput[a](3.4,-1.8){$\vec{u}\!+\!\vec{v}$} \end{pspicture}")
 \psline[arrows=->,arrows... ...v}$} \rput[a](0.3,-1.6){$\vec{v}$} \rput[a](3.4,-3.8){$\vec{u}$} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/10244/l2h/img31.png "\begin{pspicture}(0,0)(7,-5) \psdot[linecolor=red](0,0) \psline[arrows=->,arrows... ...v}$} \rput[a](0.3,-1.6){$\vec{v}$} \rput[a](3.4,-3.8){$\vec{u}$} \end{pspicture}")
 \psline[arrows=->,arrow... ...{$C$} \rput[a](9,-1.2){$\vec{m} = \frac{1}{2}(\vec{u}+\vec{v})$} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/10244/l2h/img39.png "\begin{pspicture}(0,0)(10,-5) \psdot[linecolor=red](0,0) \psline[arrows=->,arrow... ...{$C$} \rput[a](9,-1.2){$\vec{m} = \frac{1}{2}(\vec{u}+\vec{v})$} \end{pspicture}")