In space as well, for the position vectors →a, →b, and →c of A, B, and C, respectively,
- Point P that internally divides AB in the ratio m:n: →p = (n→a + m→b)/(m+n)
- Midpoint M of AB: →m = (→a + →b)/2
- Centroid G of triangle ABC: →g = (→a + →b + →c)/3
Example ① Find the midpoint of A(1,2,3) and B(4,−1,0), and the point that internally divides AB in the ratio 2:1.
Solution: Midpoint M = (5/2, 1/2, 3/2)
Internal division 2:1, P = ((1·1+2·4)/3, (1·2+2·(−1))/3, (1·3+2·0)/3) = (3, 0, 1)
Solution: Midpoint M = (5/2, 1/2, 3/2)
Internal division 2:1, P = ((1·1+2·4)/3, (1·2+2·(−1))/3, (1·3+2·0)/3) = (3, 0, 1)
Key Points
- The same formulas as in a plane hold
Practice Problems
- Point that internally divides A(0,1,2) and B(6,4,−1) in the ratio 1:2.
Answers and Explanations
- Answer: (2, 2, 1)