Question

If a, b, c are position vectors of points A, B, C respectively, with 2a + 3b -5c = 0 , then the ratio in which point C divides segment AB is

• A. 3:2 externally
• B. 2:3 externally
• C. 3:2 internally
• D. 2:3 internally

Answer 1

The Correct Option is C

Concept:
When a point divides a line segment internally in a given ratio, its position vector can be found using the section formula. If point C lies on the line segment joining points A and B, and divides it internally in the ratio m:n, then the position vector of point C is given by:
C = \(\frac{n\vec{A} + m\vec{B}}{m + n}\)

Given Equation:
2a + 3b − 5c = 0

Step 1: Rearranging the equation to isolate c:
5c = 2a + 3b
⇒ c = \(\frac{2a + 3b}{5}\)

Step 2: Now compare this with the internal division formula:
c = \(\frac{n\vec{A} + m\vec{B}}{m + n}\)
We observe:
Numerator = 2a + 3b → implies A is associated with 2, B is associated with 3
Denominator = 5 = 2 + 3

So, the ratio in which point c divides the line segment AB internally is:
AB : 3:2 (from B to A)

Conclusion:
The point c divides the line joining points A and B internally in the ratio 3:2.
Hence, the correct option is (C) 3:2 internally.