We formulate the position vectors of points that internally divide a line segment and of midpoints. These formulas make coordinate calculations much easier.
Formula for an Internal Division Point
For two points A(→a) and B(→b), the position vector →p of the point P that internally divides line segment AB in the ratio m:n is
→p = (n→a + m→b) / (m + n)
How to remember it: “the ratio to the farther endpoint × the vector of the farther endpoint”
P is n units toward A and m units toward B, so the weights of A and B are n and m. The weights cross over.
Midpoint
This is the special case m=n=1. The position vector of the midpoint M of line segment AB is
→m = (→a + →b) / 2
When A(1, 2) and B(7, 5), find the coordinates of the point P that internally divides AB in the ratio 2:1 and the midpoint M.
Solution: P：(1×→a + 2×→b)/3 = ((1+14)/3, (2+10)/3) = (5, 4)
M：((1+7)/2, (2+5)/2) = (4, 7/2)
P(5, 4)、M(4, 7/2)。
Centroid of a Triangle
If the position vectors of the vertices of triangle ABC are →a, →b, and →c, the position vector of its centroid G is
→g = (→a + →b + →c) / 3
Find the centroid G of A(0, 1), B(3, −2), and C(6, 4).
Solution: G = ((0+3+6)/3, (1−2+4)/3) = (3, 1)
- Internal division point: (n→a + m→b)/(m+n) ← the weights cross over
- Midpoint: (→a + →b)/2
- Centroid: the average of the position vectors of the three vertices
Practice Problems
- When A(2, −1) and B(5, 8), find the point P that divides AB internally in the ratio 1:2 and the midpoint M.
- Find the centroid of triangle ABC with A(1, 2), B(4, −1), and C(7, 5).
- In triangle OAB, let →OA=→a and →OB=→b. Express →OP for the point P that internally divides side AB in the ratio 3:2.
Answers and Explanations
- Answer: P(3, 2)、M(7/2, 7/2)
Explanation: P=(2×A+1×B)/3=((4+5)/3,(−2+8)/3)=(3, 2)。M=((2+5)/2,(−1+8)/2)=(7/2, 7/2)。 - Answer: (4, 2)
Explanation: (1+4+7)/3=4、(2−1+5)/3=2。 - Answer: →OP=(2→a+3→b)/5
Explanation: For internal division in the ratio 3:2, the weights cross over: 2 for A and 3 for B.