centroid

[bugzpodder]

I have a question which asks me to work with centroids for a triangle. basically i need to determine (as a function of the angle or side) what masses to place on the vertices of an triangle so that the median coincides with the orthocenter and circumcenter.

i was given this to work with:

The centroid, G of a set of points is defined vectorially by:

                              -->
              -->   Sum(m_i * OM_i)
              OG =---------------------       
                      Sum(m_i)

which looks very similar to a formula of weighted means. but i have never worked with vectors before. anyone care to give me an example?

[krtxmrtz]

I don't know exactly what kind of help you need. If you just don't feel like reading through a tedious tutorial, here's a brief example of vector addition (see figure).



Add vector OB to vector OA: graphically, shift OB parallel to itself until its origin lies on the tip of OA. Then join the origin with the tip of OB and that's your resulting vector, (you could do it the other way around and shift OA).

When you have more than 2 vectors to add, just apply this recipe recursively.

Algebraically, if you've got a sum of n vectors, call them v_i, i=1,...,n with components x_i,y_i then, the comnponents of the sum s_x, s_y are:

s_x=x₁ + x₂ + ... + x_n

s_y=y₁ + y₂ + ... + y_n

[bugzpodder]

i have no problem with vector addition/subtraction/components. if you look at the formula, it has
---->
m_i*M_i

is that plain scalar product or momentum?

[krtxmrtz]

It is just the product of a scalar (the mass) by a vector (position), it only affects its modulus. Notice the result must be a vector.

[bugzpodder]

thx for the help