vector

(ai) PQ = q - p
PR = k PQ = k(q - p)
so OR = OP + PR = p + k (q - p) = (1 - k)p + kq.
(aii) OP dot OR = p dot [ (1 - k)p + kq] = (1 - k)P^2 + kT......(1) where P = |p| and T= p dot q.
OQ dot OR = q dot [(1 - k)p + kq] = (1 - k)T + kQ^2......(2) where Q = |q|
We also know that OP dot OR = |p||r| cos θ and OQ dot OR = |q||r| cos θ,
So (OP dot OR)/(OQ dot R) = |p|/|q| = P/Q, that means (1)/(2) = P/Q
That is
[(1 - k)P^2 + kT]/[(1 - k)T + kQ^2] = P/Q
Q[(1 - k)P^2 + kT] = P[(1 - k)T + kQ^2]
QP^2 - kQP^2 + kQT = PT - kPT + kPQ^2
kQT + kPT - kQP^2 - kPQ^2 = PT - QP^2
kT(Q + P) - kPQ(P + Q) = P(T - PQ)
k(P + Q)(T - PQ) =P(T - PQ)
so k = P/(P + Q) = |p|/{ |p| + |q|}
Note : This part is much much simpler if using Sine Rule of triangle.

2012-08-10 16:09:47 補充：
(b) |p| = 13, |q| = 5, so k = 13/18, so OR = (1 - 13/18)p + 13/18q = (5p + 13q)/18
= ( - 25i + 60j + 52i + 39j)/18 = (27i + 99j)/18 = (3i + 11j)/2.
so unit vector of OR = (3i + 11j)/(2 sqrt130)