Question

If \( P(\vec{p}) \), \( Q(\vec{q}) \), \( R(\vec{r}) \), and \( S(\vec{s}) \) are four points such that \( 3\vec{p} + 8\vec{q} = 6\vec{r} + 5\vec{s} \), then the lines PQ and RS are

• A. skew
• B. intersecting
• C. parallel
• D. None of these

Hint:

If $P(\vecp)$, $Q(\vecq)$, $R(\vecr)$ and $S(\vecs)$ be four points such that $3\vecp+8\vecq=6\vecr+5\vecs$, then the lines PQ and RS are

Answer 1

The Correct Option is B

Step 1: Concept
Rewrite the equation to represent a common point using the section formula.
Step 2: Analysis
$3\vec{p} + 8\vec{q} = 6\vec{r} + 5\vec{s}$ can be written as $\frac{3\vec{p} + 8\vec{q}}{11} = \frac{6\vec{r} + 5\vec{s}}{11}$.
Step 3: Evaluation
The LHS represents a point dividing PQ in the ratio $8:3$, and the RHS represents a point dividing RS in the ratio $5:6$.
Step 4: Conclusion
Since both ratios result in the same position vector, the lines PQ and RS must intersect at that point.
Final Answer: (b)