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

p = 6i -3j +9k

q = 3i -6j -3k

r = 2p +q

r = 5i -4j +5k

R(5, -4, 5)

Explanation:

Given points P(6, -3, 9) and Q(3, -6, -3) and R that divides PQ in the ratio 1:2, you want ...

• position vectors p and q
• r in terms of p and q
• r in terms i, j, and k
• the coordinates of R

Position vector

The vector from the origin to a point (x, y, z) will be ...

xi +yj +zk

where i, j, k are unit vectors in the direction of the x, y, and z axes, respectively.

Using this pattern, we find the position vectors p and q to points P and Q to be ...

p = 6i -3j +9k . . . . . . . . . . . . . position vector p

q = 3i -6j -3k . . . . . . . . . . . . . position vector q

Division

Point R divides PQ in the ratio 1:2, so its position vector will be the weighted sum of p and q:

r = (2p +q)/3 . . . . . . . . . . . . . . . . r in terms of p and q

Using the values of p and q from above, we find r to be ...

r = (2(6i -3j +9k) +(3i -6j -3k))/3 = ((2·6+3)i +(2(-3)-6)j +(2·9-3)k)/3

r = 5i -4j +5k . . . . . . . . . . r in terms of i, j, k

Coordinates

The coordinates of R are the coordinates of the head of position vector r:

R(5, -4, 5) . . . . . . . . . . . . . coordinates of R