Centroids and Position vectors I'm currently struggling how to visualise this question. I.e where the vectors fall along the plane and how they are connected. If possible, a diagram would be great. > Let K, L, M, N be four points with position vectors k, l, m, n respectively. Write down, in terms of **k, l, m, n,** the position vectors > > **c** of point C which is the centroid of the trangle **KLM** ; > > **a** of point A which is on CN, four sevenths of the distance from **C to N**. > > Show that the vector **AK** is a multiple of the vector **AL + AM + 4AN.** Thanks so much to any replies :)

Ok, I'll do your homework. First: $$c ={1\over 3}(k+l+m)$$

$\vec{CA} = {4\over 7}\vec{CN}$ so $7a-7c =4n-4c$ and thus $$7a =3c+4n \Longrightarrow a= {1\over 7}(k+l+m+4n)$$

so finally we have:

\begin{eqnarray}\vec{AL} + \vec{AM} + 4\vec{AN} &=& l+m+4n-6a \\\&=& {1\over 7}(7l+7m+28n-6k-6l-6m-24n)\\\ &=& {1\over 7}(l+m+4n-6k)\\\ &=& {1\over 7}(l+m+4n+k)-k \\\ &=& a-k \\\ &=&-\vec{AK} \end{eqnarray}