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Directed Line Segment — Definition, Formula & Examples

A directed line segment is a line segment that runs from one specific point (the initial point) to another (the terminal point), so direction matters. It is the foundation for partition problems where you divide a segment in a given ratio.

Given two distinct points AA and BB in Rn\mathbb{R}^n, the directed line segment from AA to BB, denoted AB→\overrightarrow{AB}, is the set of points on segment AB‾\overline{AB} together with the assignment of AA as the initial point and BB as the terminal point. The direction from AA to BB is distinguished from the direction from BB to AA.

Key Formula

P=(x1+mm+n(x2−x1),  y1+mm+n(y2−y1))P = \left( x_1 + \frac{m}{m+n}(x_2 - x_1),\; y_1 + \frac{m}{m+n}(y_2 - y_1) \right)
Where:
  • (x1,y1)(x_1, y_1) = Coordinates of the initial point A
  • (x2,y2)(x_2, y_2) = Coordinates of the terminal point B
  • m:nm:n = Ratio in which P divides the segment from A to B

How It Works

When a problem says "partition directed line segment AB→\overrightarrow{AB} in the ratio m:nm:n," you find the point PP that is mm+n\frac{m}{m+n} of the way from AA toward BB. Because the segment is directed, the ratio is measured starting at AA, so AP:PB=m:nAP:PB = m:n. You apply this fraction to each coordinate separately using the section formula. Reversing the direction (starting at BB) would place the partition point in a different location, which is why direction matters.

Worked Example

Problem: Find the point P that partitions the directed line segment from A(1, 2) to B(9, 6) in the ratio 3:1.
Find the fraction: The ratio is 3:1, so the fraction of the way from A to B is
mm+n=33+1=34\frac{m}{m+n} = \frac{3}{3+1} = \frac{3}{4}
Apply to the x-coordinate: Add three-fourths of the horizontal change to the x-coordinate of A.
xP=1+34(9−1)=1+6=7x_P = 1 + \frac{3}{4}(9 - 1) = 1 + 6 = 7
Apply to the y-coordinate: Do the same for y.
yP=2+34(6−2)=2+3=5y_P = 2 + \frac{3}{4}(6 - 2) = 2 + 3 = 5
Answer: The partition point is P(7,5)P(7, 5).

Why It Matters

Partition problems on directed line segments appear frequently on state assessments and the SAT, falling under standard G-GPE.6. The same concept underlies vector notation in physics and engineering, where direction and magnitude both carry meaning.

Common Mistakes

Mistake: Ignoring the direction and computing the ratio from the wrong endpoint.
Correction: Always measure the ratio starting from the initial point of the directed segment. Partitioning AB→\overrightarrow{AB} in ratio 1:3 gives a point near A, while partitioning BA→\overrightarrow{BA} in ratio 1:3 gives a point near B.

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