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}