Set of New Years Eve

Sum of three vectors of equal length

The set of Sylvester (also formula Sylvester , named after James Joseph Sylvester ) describes a geometric interpretation of the sum of three pairs of different but of equal length vectors. Formulated as a task, it is also referred to in the literature as the problem of New Year's Eve or the triangular task of New Year's Eve .

statement

If one takes three vectors of the same length and in pairs , and from a common point and thus obtains three points , and , then the connection vector from from the point to the vertical intersection of the triangle corresponds to the sum of the three vectors, i.e.: ${\displaystyle {\vec {u}}}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {w}}}$ ${\displaystyle O}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle {\overrightarrow {OH}}}$ ${\displaystyle O}$ ${\displaystyle H}$ ${\displaystyle \triangle ABC}$

${\displaystyle {\overrightarrow {OH}}={\vec {u}}+{\vec {v}}+{\vec {w}}}$

Due to the construction of the triangle , the point is the center of the associated circumference, therefore the points and lie on the Euler line and the following relationship exists with the center of gravity of the triangle: ${\displaystyle \triangle ABC}$ ${\displaystyle O}$ ${\displaystyle O}$ ${\displaystyle H}$ ${\displaystyle S}$

${\displaystyle {\overrightarrow {OH}}=3\cdot {\overrightarrow {OS}}}$

generalization

Sum of three vectors

If one renounces the same length of the vectors and considers any three vectors different in pairs, the above relation is no longer fulfilled, but the relation to the center of gravity still applies, that is:

${\displaystyle 3\cdot {\overrightarrow {OS}}={\vec {u}}+{\vec {v}}+{\vec {w}}}$

This follows directly from the definition ${\displaystyle \mathbb {R} ^{n}}$ of the centroid for points in and the fact that in the case of the triangle, the centroid of the corners of the triangle coincide with the centroid of the triangle. Accordingly, the same applies more generally to vectors in the plane that are different in pairs and that of one common point are removed: ${\displaystyle n=3}$ ${\displaystyle n}$ ${\displaystyle O}$

${\displaystyle n\cdot {\overrightarrow {OS}}=\sum _{i=1}^{n}v_{i}}$

The focus is then on the corners of the polygon spanned by the vectors. Note that for any polygon, the center of gravity of its corners does not have to coincide with its centroid. ${\displaystyle S}$

literature

• Michael de Villiers: Generalizing a problem of New Year's Eve . In: The Mathematical Gazette , Vol. 96, No. 535 (March 2012), pp. 78-81 ( JSTOR )
• Roger A. Johnson : Advanced Euclidean Geometry . Dover 2007, ISBN 978-0-486-46237-0 , p. 251 (first published in 1929 by the Houghton Mifflin Company (Boston) under the title Modern Geometry )
• Heinrich Dörrie : 100 Great Problems of Elementary Mathematics . Dover, 1965, ISBN 0486-61348-8 , p. 142 ( online copy in the internet archive )

Web links

• Eric W. Weisstein : Sylvester's Triangle Problem . In: MathWorld (English).
• Darij Grinberg: Solution to American Mathematical Monthly Problem 11398 by Stanley Huang - contains Sylvester's theorem and proof as a subsidiary theorem