Let the points A(1,2,3), B(5,0,7) and C(−1,4,2) be the vertices of a triangle.
(i) Find the unit vector along the median from A to side BC.
(ii) Determine the length of this median.
(iii) The point D divides the median from A to BC in the ratio 2:1 (measured from A). Using the unit vector found in part (i), find the coordinates of D.
(iv) Show that the centroid of triangle ABC lies two–thirds of the way from A towards D, confirming your result in part (iii).