Relative to a fixed origin, points P, Q and R have position vectors p, q and r respectively - Edexcel - A-Level Maths Pure - Question 3 - 2020 - Paper 2

To express the relationship between the position vectors p, q, and r, we start by recognizing that since points P, Q, and R are collinear, we can represent the position vector of Q as a linear combination of the position vectors of P and R. Let us introduce a parameter t such that:

$Q = (1 - t)P + tR$

Substituting the specific condition that Q lies one third of the way from P to R, we have:

$t = \frac{1}{3}$

Thus, we write:

$Q = \left(1 - \frac{1}{3}\right)P + \frac{1}{3}R = \frac{2}{3}P + \frac{1}{3}R$