Question

Two points have the coordinates P (6, -3,9) and Q (3, -6, -3). A point R divides line PQ internally in the ratio 1:2. The position vectors of P, Q and Rare p, q and r respectively.
(a) Express the position vector of p and q.
i. In terms of p and q.
ii. In terms of i, j and k
(b) Hence state the coordinates of R.

Answer 1

(a) i. See comment...

(a) ii. We have

$\vec p = 6\,\vec\imath - 3\,\vec\jmath + 9\,\vec k$

and

$\vec q = 3\,\vec\imath - 6\,\vec\jmath - 3\,\vec k$

(b) We can parameterize the line segment from P to Q by the vector function

$\vec r(t) = (1-t) \vec p + t \vec q = (6-3t) \, \vec\imath + (-3-3t) \,\vec\jmath + (9-6t) \,\vec k$

with $0\le t\le1$. The point R is located 1/3 of the way along this line segment, i.e. at $t=\frac13$, so its position vector is

$\vec r\left(\dfrac13\right) = 5\,\vec\imath -4\,\vec\jmath + 7\,\vec k$

and hence its coordinates are $\boxed{(5,-4,7)}$.