Sum of arbitrary vertex to midpoint vectors

In summary, the conversation discussed a homework question about proving that the vectors from the vertices of a triangle to the midpoint of the opposite edge sum to zero. It was noted that this property can be extended to a set of points by forming a set of vectors using each vertex and midpoint once. Further elaboration on this proof was also provided.
  • #1
Joffan
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I was looking at a homework question posted here requiring proof that the vectors from the vertices of a triangle to the midpoint of the opposite edge sum to zero, and it struck me that there is a more general property:

Consider a set of points, [itex]\{A_0, A_1, \ldots A_n\}[/itex]. The midpoint of ##A_k## and ##A_{k+1}## is denoted by ##a_k##, with ##a_n## as the midpoint of ##A_n## and ##A_0##.

Now form a set of ##n## vectors defined by [itex]\stackrel{\rightarrow}{A_ra_s}[/itex] such that each vertex and each midpoint is used once by a vector in the set.

Now that set of vectors will sum to zero.

--------------------------

Sketch of messy proof
First step - the midpoints are a distraction. ##\stackrel{\rightarrow}{A_ra_s} = \stackrel{\rightarrow}{A_rA_s} + \stackrel{\rightarrow}{A_sa_s} = \stackrel{\rightarrow}{A_rA_s }+ \frac{1}2\stackrel{\rightarrow}{A_sA_{s+1}}##, so the vector sum contains a scaled loop from the midpoint steps which sums to zero.

Next step - The remaining vectors form a set of closed loops which also sum to zero.

Improvements on this welcome...
 
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  • #2
The set of points A_k and a_k may be considered vectors from origon. So the n vectors you describe are on the form a_s-A_r, where s and r ranges from 0 to n with r depending on s via some bijection (<-- I assume this is what you mean).

a_k may be written as [itex]\frac{1}{2}(A_k+A_{k+1})[/itex], where indices are considered modulo n+1.

By writing out entire the sum, we get exactly 0.
 
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  • #3
Nice and neat. Thanks.
 

What is the sum of arbitrary vertex to midpoint vectors?

The sum of arbitrary vertex to midpoint vectors refers to the sum of all the vectors that connect a vertex to the midpoint of each of its adjacent edges in a polygon. This sum is used to calculate the centroid of the polygon.

How is the sum of arbitrary vertex to midpoint vectors calculated?

The sum of arbitrary vertex to midpoint vectors can be calculated by first finding the midpoint of each edge in the polygon. Then, a vector is drawn from each vertex to its corresponding midpoint. Finally, all these vectors are added together to find the total sum.

What is the significance of the sum of arbitrary vertex to midpoint vectors?

The sum of arbitrary vertex to midpoint vectors is used to find the centroid of a polygon. The centroid is an important point in a polygon that represents its geometric center and has various applications in mathematics, physics, and engineering.

Can the sum of arbitrary vertex to midpoint vectors be negative?

No, the sum of arbitrary vertex to midpoint vectors cannot be negative. Vectors have both magnitude and direction, but the sum of vectors only takes into account their magnitudes. Therefore, the result of the sum will always be positive or zero.

Is the sum of arbitrary vertex to midpoint vectors the same for all polygons?

No, the sum of arbitrary vertex to midpoint vectors will vary for different polygons. It depends on the number of sides and lengths of the edges of the polygon. However, the centroid of any polygon can be found using this method.

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