Let `vec(p) and vec(q)` be the position vectors of the point P and Q respectively with respect to origin O. The points R and S divide PQ internally and externally respectiely in the ratio `2:3`. If `vec(OR) and vec(OS)` are perpendicular, then which one of the following is correct?

Options

A. A) `9p^(2)= 4q^(2)`
B. B) `4p^(2)= 9q^(2)`
C. C) `9p= 4q`
D. D) `4p=9q`

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the position vectors
Let the position vectors of points P and Q be represented as:
- \( \vec{OP} = \vec{p} \)
- \( \vec{OQ} = \vec{q} \)

### Step 2: Find the position vector of point R (internal division)
Point R divides the segment PQ internally in the ratio 2:3. According to the section formula for internal division, the position vector of R is given by:
\[ \vec{OR} = \frac{n \vec{p} + m \vec{q}}{n + m} \]
where \( m = 3 \) and \( n = 2 \). Thus,
\[ \vec{OR} = \frac{3\vec{p} + 2\vec{q}}{3 + 2} = \frac{3\vec{p} + 2\vec{q}}{5} \]

### Step 3: Find the position vector of point S (external division)
Point S divides the segment PQ externally in the ratio 2:3. According to the section formula for external division, the position vector of S is given by:
\[ \vec{OS} = \frac{n \vec{p} - m \vec{q}}{n - m} \]
where \( m = 3 \) and \( n = 2 \). Thus,
\[ \vec{OS} = \frac{3\vec{p} - 2\vec{q}}{3 - 2} = 3\vec{p} - 2\vec{q} \]

### Step 4: Set up the condition for perpendicular vectors
Since \( \vec{OR} \) and \( \vec{OS} \) are perpendicular, their dot product must equal zero:
\[ \vec{OR} \cdot \vec{OS} = 0 \]

### Step 5: Substitute the expressions for \( \vec{OR} \) and \( \vec{OS} \)
Substituting the expressions we found:
\[ \left( \frac{3\vec{p} + 2\vec{q}}{5} \right) \cdot (3\vec{p} - 2\vec{q}) = 0 \]

### Step 6: Simplify the dot product
Expanding the dot product:
\[ \frac{1}{5} \left( (3\vec{p} + 2\vec{q}) \cdot (3\vec{p}) - (3\vec{p} + 2\vec{q}) \cdot (2\vec{q}) \right) = 0 \]
This simplifies to:
\[ \frac{1}{5} \left( 9\vec{p} \cdot \vec{p} + 6\vec{q} \cdot \vec{p} - 6\vec{p} \cdot \vec{q} - 4\vec{q} \cdot \vec{q} \right) = 0 \]
This leads to:
\[ 9\vec{p}^2 + 6\vec{q} \cdot \vec{p} - 6\vec{p} \cdot \vec{q} - 4\vec{q}^2 = 0 \]
Since \( \vec{q} \cdot \vec{p} = \vec{p} \cdot \vec{q} \), we can simplify this to:
\[ 9\vec{p}^2 - 4\vec{q}^2 = 0 \]

### Step 7: Conclusion
From the equation \( 9\vec{p}^2 = 4\vec{q}^2 \), we can conclude that the relationship between the magnitudes of the vectors is established.