Question

If the position vectors of the points P and Q are, respectively, \(5\vec{a} - 6\vec{b}\) and \(\vec{a} + 2\vec{b}\), then the point R with position vector \(2\vec{a}\) divides the line segment joining P and Q internally in the ratio

• A. 3:2
• B. 3:1
• C. 2:1
• D. 2:3
• E. 3:4

Hint:

To save time, only equate the coefficients of one vector (either \(\vec{a}\) or \(\vec{b}\)). If the division is internal, you should get a positive ratio. If you get a negative value, the division is external.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The section formula for internal division states that if a point R with position vector \(\vec{r}\) divides the line joining points P (\(\vec{p}\)) and Q (\(\vec{q}\)) in the ratio \(m:n\), then: \[ \vec{r} = \frac{m\vec{q} + n\vec{p}}{m+n} \]

Step 2: Key Formula or Approach:
Substitute the given vectors \(\vec{p} = 5\vec{a} - 6\vec{b}\), \(\vec{q} = \vec{a} + 2\vec{b}\), and \(\vec{r} = 2\vec{a}\) into the section formula and solve for the ratio \(m/n\).

Step 3: Detailed Explanation:
Assume the ratio is \(m:n\). Then: \[ 2\vec{a} = \frac{m(\vec{a} + 2\vec{b}) + n(5\vec{a} - 6\vec{b})}{m+n} \]
Multiply both sides by \((m+n)\): \[ (2m + 2n)\vec{a} = (m + 5n)\vec{a} + (2m - 6n)\vec{b} \]
Equating the coefficients of \(\vec{a}\) and \(\vec{b}\) on both sides: 1. For \(\vec{b}\): Since the left side has no \(\vec{b}\) component, the coefficient must be zero: \[ 2m - 6n = 0 \implies 2m = 6n \implies \frac{m}{n} = \frac{6}{2} = \frac{3}{1} \]
2. For \(\vec{a}\): \[ 2m + 2n = m + 5n \implies m = 3n \implies \frac{m}{n} = \frac{3}{1} \]
Both equations yield the same ratio \(3:1\).

Step 4: Final Answer:
The point R divides PQ in the ratio 3:1.