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'parallelogram principle'
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| Title of object: |
parallelogram principle |
| Canonical Name: |
ParallelogramPrinciple |
| Type: |
Topic |
| Created on: |
2008-02-06 16:39:48 |
| Modified on: |
2008-02-06 17:59:32 |
| Classification: |
msc:53A45 |
| Defines: |
sum of vectors, sum |
| Synonyms: |
parallelogram principle=addition of vectors
parallelogram principle=vector addition |
Revision comment (for changes between this and next version):
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
\usepackage{pstricks}
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
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Content:
\PMlinkescapeword{right} \PMlinkescapeword{opposite sides}
A starting \PMlinkescapetext{point} for learning vectors is to think that they are {\em directed line segments}. Thus a vector $\vec{u}$ has a direction and a length (magnitude) and nothing else. Therefore, if two vectors $\vec{u}$ and $\vec{v}$ have a same direction and a same length, one can keep them identical (and denote\, $\vec{u} = \vec{v}$). 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 $\vec{u}$ as an infinite set of directed segments is quite uncomfortable to handle, and one can choose from all possible representants of $\vec{u}$ one individual directed segment $\overrightarrow{AB}$, i.e. a line segment directed from a certain point $A$ (the {\em initial point}) to another certain point $B$ (the {\em terminal point}). Although $\overrightarrow{AB}$ is only a representant of $\vec{u}$, one may write\, $\overrightarrow{AB} = \vec{u}$\, or\, $\vec{u} = \overrightarrow{AB}$.
For describing a vector $\vec{u}$, 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 $\vec{u}$ means from left to right (i.e. in the direction of $x$-axis) and how great from below upwards (i.e. in the direction of the $y$-axis); those displacements may be expressed with two numbers.\, One may for example write
\begin{align}
\vec{u} = \left(\!\begin{array}{c} +5\\-1 \end{array}\!\right)\!,
\end{align}
where the first (upper) number $+5$ tells that the vector leads 5 length-units to the right and the second (lower) number $-1$ that it leads 1 length-unit downwards.
Then to the {\em addition of vectors}!\, Since the vector may be interpreted as a \PMlinkescapetext{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
\begin{align}
\vec{v} = \left(\!\begin{array}{c} +1\\-3 \end{array}\!\right)\!,
\end{align}
then the {\em sum} of the vectors (1) and (2) is
$$\vec{u}+\vec{v} =
\left(\!\begin{array}{c} +5\\-1 \end{array}\!\right)+\left(\!\begin{array}{c} +1\\-3 \end{array}\!\right) =
\left(\!\begin{array}{c} +5+1\\-1-3 \end{array}\!\right) = \left(\!\begin{array}{c} +6\\-4 \end{array}\!\right)\!,$$
which result means a vector leading 6 length-units to the right and 4 down.
\begin{center}
\begin{pspicture}(0,0)(7,-5)
\psdot[linecolor=red](0,0)
\psline[linecolor=green]{->}(0,0)(5,-1)
\psline[linecolor=cyan]{->}(5,-1)(6,-4)
\psline[linecolor=blue]{->}(0,0)(6,-4)
\rput[a](2.8,-0.3){$\vec{u}$}
\rput[a](5.7,-2.2){$\vec{v}$}
\rput[a](3.4,-1.8){$\vec{u}\!+\!\vec{v}$}
\end{pspicture}
\end{center}
When we set the vectors $\vec{u}$ and $\vec{v}$ one behind the other, as in the pcture, 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.
Note, that the sum vector $\vec{u}\!+\!\vec{v}$ can be also obtained as the diagonal vector of the parallelogram with one pair of opposite sides equal to $\vec{u}$ and the other pair of opposite sides equal to $\vec{v}$.
\begin{center}
\begin{pspicture}(0,0)(7,-5)
\psdot[linecolor=red](0,0)
\psline[linecolor=green]{->}(0,0)(5,-1)
\psline[linecolor=cyan]{->}(5,-1)(6,-4)
\psline[linecolor=cyan]{->}(0,0)(1,-3)
\psline[linecolor=green]{->}(1,-3)(6,-4)
\psline[linecolor=blue]{->}(0,0)(6,-4)
\rput[a](2.8,-0.3){$\vec{u}$}
\rput[a](5.7,-2.2){$\vec{v}$}
\rput[a](3.4,-1.8){$\vec{u}\!+\!\vec{v}$}
\rput[a](0.3,-1.6){$\vec{v}$}
\rput[a](3.4,-3.8){$\vec{u}$}
\end{pspicture}
\end{center}
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