# parallelogram principle

Since the vector may be interpreted as a 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

 $\displaystyle\vec{v}=\left(\!\begin{array}[]{c}+1\\ -3\end{array}\!\right)\!,$ (2)

then the 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.

When we set the vectors $\vec{u}$ and $\vec{v}$ 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

 $\overrightarrow{PQ}+\overrightarrow{QR}=\overrightarrow{PR}.\\$

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 $\vec{v}$.  The parallelogram picture illustrates also that the vector addition is commutative, i.e. that  $\vec{u}+\vec{v}=\vec{v}+\vec{u}$.

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 $\vec{u}+\vec{v}$ is halved into two equal vectors (better: directed segments) $\vec{m}=\frac{1}{2}(\vec{u}+\vec{v})$.  In the triangle  $ABC$, the vectors $\vec{u}$, $\vec{v}$, $\vec{m}$ may be called two side vectors and a median vector, all having the common initial point $A$.  Thus we can write the

In a triangle, the median vector emanating from a certain vertex is the arithmetic mean  of the side vectors emanating from the same vertex.

 Title parallelogram principle Canonical name ParallelogramPrinciple Date of creation 2013-03-22 17:47:04 Last modified on 2013-03-22 17:47:04 Owner pahio (2872) Last modified by pahio (2872) Numerical id 16 Author pahio (2872) Entry type Topic Classification msc 53A45 Synonym addition of vectors Synonym vector addition Synonym sum vector Related topic CommonPointOfTriangleMedians Related topic ProvingThalesTheoremWithVectors Defines directed line segment Defines sum of vectors Defines side vector Defines median vector Defines diagonal vector