Where Two Service Paths Cross
A civil engineer is laying two service paths across a triangular development plot OPQ. Relative to a survey marker at O, corner P has position vector p and corner Q has position vector q, with p and q not parallel. Path 1 runs from O to a junction R on the boundary PQ, where PR : RQ = 2 : 1. Path 2 runs from P to a junction S on OQ, where OS : SQ = 1 : 1. The paths cross at T, where a lamp post must be installed. The engineer needs the ratio OT : TR to choose the lamp wattage, and PT : TS to order the right length of cable from P. A junior technician objects that the problem cannot be solved because no lengths or angles have been supplied - only the two letters p and q - and proposes going back to site with a tape measure. The available drawing shows the plot is neither right-angled nor isosceles.
1. Express OR and PS in terms of p and q. Explain why the technician's objection is wrong, and state the condition on p and q that your method requires.
Read the worked answer
Finding OR
R divides PQ in the ratio PR : RQ = 2 : 1, so R is two-thirds of the way from P to Q:
OR = OP + (2/3)PQ = p + (2/3)(q − p) = (1/3)p + (2/3)q.
Finding PS
S is the midpoint of OQ, so OS = ½q and
PS = OS − OP = ½q − p = −p + ½q.
Why the objection is wrong
A vector equation does not need numerical lengths. If p and q are non-parallel, they act as a basis: every position vector in the plane can be written in exactly one way as αp + βq. Ratios of division along a line are therefore fixed by the coefficients alone, whatever the actual lengths and angles of the plot - the answer is shape-independent.
The method requires p and q to be non-zero and not parallel; otherwise the representation αp + βq is not unique (the two basis directions collapse onto one line) and equating coefficients would be invalid.
2. Given that OT = λOR and OT = OP + μPS, find the values of λ and μ.
Read the worked answer
First route to T (along OR)
OT = λ[(1/3)p + (2/3)q] = (λ/3)p + (2λ/3)q
Second route to T (along PS)
OT = p + μ(−p + ½q) = (1 − μ)p + (μ/2)q
Equate coefficients of p and q (valid because p and q are non-parallel):
λ/3 = 1 − μ … (1)
2λ/3 = μ/2 … (2)
From (2): μ = 4λ/3. Substituting into (1):
λ/3 = 1 − 4λ/3
λ/3 + 4λ/3 = 1
5λ/3 = 1 → λ = 3/5, and μ = 4(3/5)/3 = 4/5.
Check
OT = (1/5)p + (2/5)q from the first route, and (1 − 4/5)p + (2/5)q = (1/5)p + (2/5)q from the second. Consistent.
3. Hence state OT : TR and PT : TS, and verify that your point T lies on both paths.
Read the worked answer
OT : TR
OT = (3/5)OR means T is three-fifths of the way along OR, so OT : TR = 3 : 2.
PT : TS
PT = μPS with μ = 4/5, so PT : TS = 4 : 1.
Position of T
T = (1/5)p + (2/5)q.
On path 1: T = (3/5)[(1/3)p + (2/3)q] = (1/5)p + (2/5)q ✓
On path 2: T = p + (4/5)(½q − p) = p − (4/5)p + (2/5)q = (1/5)p + (2/5)q ✓
Practical reading for the engineer
The lamp post stands at the point dividing OR in 3 : 2, i.e. 60% of the way from O to R - no site measurement of lengths or angles is needed. Note also that T is not the centroid of the plot, which would be (1/3)p + (1/3)q; the coefficients must not be assumed rather than derived.