Vectors - Prove the following relation about the centroid

In summary, we have a triangle OAB and a variable point P, with G being the centroid. The objective is to prove that PA2+PB2+PO2=GA2+GB2+GO2+3 (PG2). By treating P as the origin and using g = (1/3)(a + b + c), the equation can be simplified to a2+b2+c2=1/9(|a+b-2c|2+|b+c-2a|2+|a+c-2b|2+3|a+b+c|2). To prove this, start with the complicated part of the equation, (g - a)2 + (g - b)2 + (
  • #1
ritwik06
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0

Homework Statement


http://img147.imageshack.us/img147/733/vectorsba4.png [Broken]
Given is the triangle OAB and a variable point P. G is the centroid. Prove that:
PA2+PB2+PO2=GA2+GB2+GO2+3 (PG2)


The Attempt at a Solution


I treat O as the origin.
Vectors are denoted in bold.
Position vectors of A, B, P are a,b and p respectively.
G=a+b/3
How on Earth am I going to prove that:
9(|p-a|2+|p-b|2+|p|2)=|b-2a|2+|a-2b|2+|a+b|2+3|a+b-3p|2

I mean is there any criteria to prove this equation in modulus of vectors when I don't know the angles between any of them?
 
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  • #2
ritwik06 said:
Given is the triangle OAB and a variable point P. G is the centroid. Prove that:
PA2+PB2+PO2=GA2+GB2+GO2+3 (PG2)

I treat O as the origin.
Vectors are denoted in bold.
Position vectors of A, B, P are a,b and p respectively.
G=a+b/3

Hi ritwik06! :smile:

No, it'll be easier ('cos it's more symmetrical) if you treat P as the origin, and use g = (1/3)(a + b + c) :wink:

(when the answer is symmetrical, always try to keep the proof symmetrical! :wink:)
 
  • #3


tiny-tim said:
Hi ritwik06! :smile:

No, it'll be easier ('cos it's more symmetrical) if you treat P as the origin, and use g = (1/3)(a + b + c) :wink:

(when the answer is symmetrical, always try to keep the proof symmetrical! :wink:)

Thanks a lot tim.
I have done exactly that. And I get:
a2+b2+c2=1/9(|a+b-2c|2+|b+c-2a|2+|a+c-2b|2+3|a+b+c|2)

But the fact still remains that I don't know many angle such as th one made by b+c-2a.?? I think it is 0.
 
  • #4
No … start with the complicated part of the equation, and try to simplify it, not the other way round!

In other words, start with (g - a)2 + (g - b)2 + (g - c)2 :smile:
 

What is a centroid and how is it related to vectors?

The centroid is the geometric center of a shape or object. In the context of vectors, the centroid is the point where the medians of a triangle intersect. The medians of a triangle are lines that connect a vertex to the midpoint of the opposite side. Therefore, the centroid of a triangle can be found by taking the average of the coordinates of its three vertices.

How can the centroid be expressed as a vector?

The centroid of a triangle can be expressed as a vector by taking the average of the coordinates of the three vectors that represent the sides of the triangle. This means adding the x-coordinates and dividing by 3, then adding the y-coordinates and dividing by 3. The resulting vector will have the same direction as the medians and its magnitude will be equal to 1/3 of the sum of the magnitudes of the three vectors.

What is the relation between the centroid and the position vectors of the vertices?

The relation between the centroid and the position vectors of the vertices is that the centroid is located at the point where the three position vectors intersect. This means that the centroid vector can be expressed as the sum of the three position vectors, each multiplied by 1/3. In other words, the centroid vector is the average of the position vectors of the three vertices.

How can the centroid be used to prove the following relation about vectors?

The centroid can be used to prove the following relation about vectors: the sum of the squares of the distances from the centroid to the vertices is equal to 3/4 of the sum of the squares of the lengths of the sides of the triangle. This can be proven by using the definition of the centroid as the average of the position vectors and the distance formula for calculating the lengths of the sides of the triangle.

Can the relation about the centroid be extended to other polygons?

Yes, the relation about the centroid can be extended to other polygons. The centroid of any polygon can be found by taking the average of the coordinates of its vertices. Additionally, the relation about the centroid can be extended to other polygons by using the same approach as for triangles, but with different formulas for calculating the lengths of the sides and the distances from the centroid to the vertices.

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