We solve problems by combining our knowledge of internal division, external division, and centroids.
Decomposing Position Vectors
Example ①
Let P be the point that internally divides side AB of triangle OAB in the ratio 2:3, and let Q be the midpoint of side OB. Given →OA=→a and →OB=→b, express →PQ.
Solution: →OP=(3→a+2→b)/5、→OQ=→b/2。
→PQ=→OQ−→OP=(→b/2)−(3→a+2→b)/5=(−6→a − 4→b + 5→b)/10×... Taking a common denominator,
=(5→b−2(3→a+2→b))/10=(−6→a+→b)/10。 Therefore, →PQ=(−3→a+→b/2)/5 = (−6→a + →b)/10。
Let P be the point that internally divides side AB of triangle OAB in the ratio 2:3, and let Q be the midpoint of side OB. Given →OA=→a and →OB=→b, express →PQ.
Solution: →OP=(3→a+2→b)/5、→OQ=→b/2。
→PQ=→OQ−→OP=(→b/2)−(3→a+2→b)/5=(−6→a − 4→b + 5→b)/10×... Taking a common denominator,
=(5→b−2(3→a+2→b))/10=(−6→a+→b)/10。 Therefore, →PQ=(−3→a+→b/2)/5 = (−6→a + →b)/10。
Three Collinear Points
Example ②
Find k such that A(2, 1), B(5, 4), and C(k, 10) are collinear.
Solution: →AB=(3, 3)、→AC=(k−2, 9)。By the parallel condition, 3×9−3(k−2)=0 → 27=3k−6 → k=11。
Find k such that A(2, 1), B(5, 4), and C(k, 10) are collinear.
Solution: →AB=(3, 3)、→AC=(k−2, 9)。By the parallel condition, 3×9−3(k−2)=0 → 27=3k−6 → k=11。
Combining Centroids and Division Points
Example ③
For triangle A(1, 0), B(4, 0), and C(2, 3), find the coordinates of the centroid G. Also find the midpoint M of AG.
Solution: G=((1+4+2)/3, (0+0+3)/3)=(7/3, 1)。
M=((1+7/3)/2, (0+1)/2)=(10/6, 1/2)=(5/3, 1/2)
For triangle A(1, 0), B(4, 0), and C(2, 3), find the coordinates of the centroid G. Also find the midpoint M of AG.
Solution: G=((1+4+2)/3, (0+0+3)/3)=(7/3, 1)。
M=((1+7/3)/2, (0+1)/2)=(10/6, 1/2)=(5/3, 1/2)
Key Points
- For figure problems, expressing everything using basic vectors →a and →b makes the structure clear
- Three points are collinear: →AB ∥ →AC
- Problems combining division-point and centroid formulas appear frequently
Practice Problems
- Find k such that A(2, −1), B(5, 2), and C(k, 8) are collinear.
- In triangle ABC with A(0, 0), B(6, 0), and C(0, 9), find the vector connecting the centroid and the midpoint of side BC.
- In triangle OAB, let →OA=→a and →OB=→b. If P internally divides OA in the ratio 1:2 and Q is the midpoint of OB, express →PQ.
Answers and Explanations
- Answer: k=11
Explanation: →AB=(3,3)、→AC=(k−2,9)。3×9−3(k−2)=0 → k=11。 - Answer: (0, 3/2)→ centroid(2, 3), midpoint(3, 9/2). Vector=(1, 3/2)
Explanation: Centroid G(2,3), midpoint M of BC(3, 9/2). →GM=(1, 3/2)。 - Answer: →PQ=−→a/3+→b/2=(−2→a+3→b)/6
Explanation: →OP=→a/3、→OQ=→b/2。→PQ=→OQ−→OP=→b/2−→a/3。