An “external division point” on the extension of a line segment can also be represented by a position vector using the same framework as internal division.
Formula for an External Division Point
For two points A(→a) and B(→b), the position vector →q of the point Q that externally divides line segment AB in the ratio m:n is
→q = (−n→a + m→b) / (m − n)
This is obtained by replacing n in the internal division formula with −n.
Meaning of External Division
- When m＞n: Q is outside B (farther from A than B)
- When m＜n: Q is outside A (on the side before A)
- When m=n: external division is undefined because the denominator is 0
Example ①
When A(1, 2) and B(4, 5), find the point Q that externally divides AB in the ratio 3:1.
Solution: Q = (−1×→a + 3×→b)/(3−1) = ((−1+12)/2, (−2+15)/2) = (11/2, 13/2)
Therefore, Q(11/2, 13/2)
When A(1, 2) and B(4, 5), find the point Q that externally divides AB in the ratio 3:1.
Solution: Q = (−1×→a + 3×→b)/(3−1) = ((−1+12)/2, (−2+15)/2) = (11/2, 13/2)
Therefore, Q(11/2, 13/2)
Geometric Interpretation of External Division
“Externally divide AB in the ratio m:n” can be interpreted as AQ:QB = m:(−n) (giving QB a weight in the opposite direction). Viewed from Q, B lies on the opposite side from A compared with an internal division point.
Example ②
When A(2, 1) and B(5, 7), find the point R that externally divides AB in the ratio 1:3.
Solution: R = (−3×→a + 1×→b)/(1−3) = ((−6+5)/(−2), (−3+7)/(−2)) = (1/2, −2)
Therefore, R(1/2, −2)
When A(2, 1) and B(5, 7), find the point R that externally divides AB in the ratio 1:3.
Solution: R = (−3×→a + 1×→b)/(1−3) = ((−6+5)/(−2), (−3+7)/(−2)) = (1/2, −2)
Therefore, R(1/2, −2)
Key Points
- External division point: (−n→a + m→b)/(m−n)
- Obtained by replacing n in the internal division formula with −n
- When m=n, external division is undefined (infinity on the extension of the segment)
Practice Problems
- When A(0, 0) and B(6, 3), find the point Q that externally divides AB in the ratio 4:1.
- When A(−1, 2) and B(3, 4), find the point R that externally divides AB in the ratio 1:2.
- In triangle OAB, let →OA=→a and →OB=→b. Express →OP for the point P that externally divides side AB in the ratio 3:1.
Answers and Explanations
- Answer: (8, 4)
Explanation: (−1×0+4×6)/(4−1)=24/3=8、(−1×0+4×3)/3=4。 - Answer: (−5, 0)
Explanation: (−2×(−1)+1×3)/(1−2)=5/(−1)=−5、(−2×2+1×4)/(−1)=0。 - Answer: →OP=(−→a+3→b)/2
Explanation: External division in the ratio 3:1 gives (−1×→a+3×→b)/(3−1).