A Vectors question from my maths class, can you solve it?

Question

ABCD is a quadrilateral with vertices A(4,-1,3), B(8,3,-1), C(0,0,4) and D(-4,0,8).

(a)Find the coordinates of M, the midpoint of AB

(b) Find the coordinates of the point T, which divides CM in the ratio 2 : 1

(c) Show that B, T and D are collinear and find the ratio in which T divides BD.

Answer 1

(a) If P = (x, y) and Q = (u, v), then the midpoint of PQ is R = ((x + u) / 2, (y + v) / 2).

(b) If S divides PQ in the ratio a : b. then S = ( (b x + a u) / (a + b), (b y + a v) / (a + b) ).

(c) Three points X, Y and Z are collinear precisely when the slopes of XY and YZ are the same. Further, supposing X-Y-Z, then Y divides XZ in the ratio |XY| : |YZ|. So to solve part (c), you must know how to calculate the slopes and lengths of XY, given the points X and Y.