9 min readConcept sheet1-mark MCQ or assertion-reason; 2-mark and 3-mark questions; often part of a case-based question

Position vector of a point dividing a segment

If R divides PQ internally in the ratio m : n, then r⃗ = (mb⃗ + na⃗)/(m + n). Here a⃗, b⃗ are the position vectors of P, Q.

Internal division (PR : RQ = m : n)r⃗ = (mb⃗ + na⃗)/(m + n)

External divisionr⃗ = (mb⃗ − na⃗)/(m − n), m ≠ n

Midpointr⃗ = (a⃗ + b⃗)/2

Pairing rulem goes with the far end Q (b⃗); n goes with the near end P (a⃗)

How to draw the figure

1
Draw a line segment PQ and mark R on it, between P and Q.
2
Write m above PR and n above RQ.
3
Mark the origin O away from the line. Draw arrows from O to P, R and Q.
4
Label the arrows a⃗ (to P), r⃗ (to R) and b⃗ (to Q).

Derivation (internal division)

1
Let OP⃗ = a⃗, OQ⃗ = b⃗, OR⃗ = r⃗, with R on PQ and PR : RQ = m : n.
2
PR⃗ and RQ⃗ point the same way, so n PR⃗ = m RQ⃗.
3
PR⃗ = r⃗ − a⃗ and RQ⃗ = b⃗ − r⃗.
4
n(r⃗ − a⃗) = m(b⃗ − r⃗)
5
nr⃗ + mr⃗ = mb⃗ + na⃗
6
So, r⃗ = (mb⃗ + na⃗)/(m + n).

External division and the midpoint

R divides PQ externally in the ratio m : n when R lies on line PQ outside the segment and PR : QR = m : n. Then n PR⃗ = m QR⃗, so n(r⃗ − a⃗) = m(r⃗ − b⃗), giving r⃗ = (mb⃗ − na⃗)/(m − n). For the midpoint, put m = n = 1 in the internal formula: r⃗ = (a⃗ + b⃗)/2. External division in the ratio 1 : 1 is impossible, since m − n = 0. Extra help (not in your textbook): the centroid G of triangle ABC divides the median from A in the ratio 2 : 1, which gives OG⃗ = (a⃗ + b⃗ + c⃗)/3.

Example 1: internal and external

Problem

P and Q have position vectors a⃗ = 2î − 3ĵ + 4k̂ and b⃗ = 5î + 3ĵ − 2k̂. Find the position vector of R that divides PQ in the ratio 2 : 1 (i) internally, (ii) externally.

Working

(i) r⃗ = (2b⃗ + 1a⃗)/(2 + 1)
2b⃗ + a⃗ = (10 + 2)î + (6 − 3)ĵ + (−4 + 4)k̂ = 12î + 3ĵ
r⃗ = (12î + 3ĵ)/3 = 4î + ĵ
(ii) r⃗ = (2b⃗ − 1a⃗)/(2 − 1)
= (10 − 2)î + (6 + 3)ĵ + (−4 − 4)k̂ = 8î + 9ĵ − 8k̂

Answer

(i) 4î + ĵ; (ii) 8î + 9ĵ − 8k̂.

Example 2: board-style, find the ratio

Problem

Find the ratio in which C(3, 1, 3) divides the segment joining A(1, −1, 5) and B(7, 5, −1).

Working

Let C divide AB in the ratio k : 1.
Then OC⃗ = (k OB⃗ + OA⃗)/(k + 1).
x-coordinate: 3 = (7k + 1)/(k + 1), so 3k + 3 = 7k + 1 and k = 1/2.
Check y: (5k − 1)/(k + 1) = (2.5 − 1)/1.5 = 1. Check z: (−k + 5)/(k + 1) = 4.5/1.5 = 3. Both match C.
k = 1/2 is positive, so the division is internal, in the ratio 1/2 : 1 = 1 : 2.

Answer

C divides AB internally in the ratio 1 : 2.

In real life

Planning a metro station one-third of the way between two stops uses the section formula. So does placing a rest stop halfway along a highway.

Exam angle

1 marks: “Find the position vector of the midpoint of the segment joining A(1, −3, 5) and B(3, 7, −1).” (a⃗ + b⃗)/2 = [(1 + 3)î + (−3 + 7)ĵ + (5 − 1)k̂]/2 = 2î + 2ĵ + 2k̂.

2 marks: “P and Q have position vectors î + ĵ and 3î − ĵ + 2k̂. Find the position vector of R that divides PQ externally in the ratio 3 : 1.” r⃗ = (3q⃗ − p⃗)/(3 − 1) = (9î − 3ĵ + 6k̂ − î − ĵ)/2 = (8î − 4ĵ + 6k̂)/2 = 4î − 2ĵ + 3k̂.

Remember it

m with the far end, n with the near end. Plus inside, minus outside.