Derive the position vector of the point R that divides the segment joining P(a⃗) and Q(b⃗) internally in the ratio m : n. Hence find the point dividing the join of 2î + ĵ − 3k̂ and −4î + 7ĵ + 3k̂ internally in the ratio 1 : 2.

1. 1 · Write what is given and what is to be found. Given: OP⃗ = a⃗, OQ⃗ = b⃗, PR : RQ = m : n. To find: r⃗ = OR⃗, then apply it.
2. 2 · Draw the figure. Segment PQ with R between, m and n marked; origin O with arrows a⃗, r⃗, b⃗ to P, R, Q.
3. 3 · Use the ratio.
   • PR⃗ and RQ⃗ point the same way, so n PR⃗ = m RQ⃗
   • n(r⃗ − a⃗) = m(b⃗ − r⃗)
4. 4 · Solve for r⃗. (m + n)r⃗ = mb⃗ + na⃗, so r⃗ = (mb⃗ + na⃗)/(m + n).
5. 5 · Apply the formula.
   • m = 1, n = 2: r⃗ = [1(−4î + 7ĵ + 3k̂) + 2(2î + ĵ − 3k̂)]/3
   • = (0î + 9ĵ − 3k̂)/3 = 3ĵ − k̂
6. 6 · Check and box the answer. r⃗ = (mb⃗ + na⃗)/(m + n); the required point has position vector 3ĵ − k̂, that is (0, 3, −1).

Scoring tip: Write n PR⃗ = m RQ⃗ before anything else. Swapping m and n is the most common slip in this derivation.