What is the Linear Combination of Vectors for the Centroid of a Triangle?

[lucfuture]

Homework Statement

http://delphi.zsg-rottenburg.de/gif/1la1_pyramide.gif
It says "M is the midpoint of BD and S is the center of triangle ABC. Express vector SM as a linear combination of A, B, and C."

The Attempt at a Solution

I think I am correct in saying that SM is half the magnitude of vector C and has the same angle with respect to the plane that triangle ABC is in.

 

[Outlined]

You mean a linear combination of a, b and c !
My attempt would be to write to point S as a lin.comb. and also the point M as a lin.comb. I would also avoid working with angles, uses techniques like averages instead.

 

[lucfuture]

You mean a linear combination of a, b and c !
My attempt would be to write to point S as a lin.comb. and also the point M as a lin.comb. I would also avoid working with angles, uses techniques like averages instead.

You are correct! sorry. so S would be 1/2a+1/2b and M would be 3^1/2/2 a+3^1/2/2 c?

 

[Outlined]

I think M is (a + c) / 2.
Not sure about S. I would say S is (0 + a + b) / 3.

Some extra reading you might like:
http://www.netcomuk.co.uk/~jenolive/centroid.html

 

[lucfuture]

Ok a agree that M is (a+c)/2. For S I would say it is (a+b)/4 because M' would be (a+b)/2 by the same method we got M. And S is half the distance that M' is from point A.

does that make sense?

 

[lucfuture]

And then you would use vector subtraction of vectors AM - AS to get the answer?

 

[Outlined]

Well, S = (0 + a + b) / 3. Also have a look at that link, the middle is called the centroid.