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

Options

A. Skew
B. intersecting
C. parallel
D. None of

Answer

To solve the problem, we need to analyze the given equation involving the vectors of points P, Q, R, and S:

Given:
\[ 3\vec{p} + 8\vec{q} = 6\vec{r} + 5\vec{s} \]

We need to determine the relationship between the lines PQ and RS.

### Step 1: Rearranging the Equation
We can rearrange the given equation to isolate the terms involving vectors:
\[ 3\vec{p} + 8\vec{q} - 6\vec{r} - 5\vec{s} = 0 \]

### Step 2: Analyzing the Equation
This equation represents a linear combination of the vectors. For the lines PQ and RS to be parallel or intersecting, we need to analyze the coefficients of the vectors.

### Step 3: Assuming Values for Vectors
Let’s assume specific values for the vectors to check the relationship. We can set:
\[ \vec{p} = \vec{q} = \vec{r} = \vec{s} = \vec{a} \quad \text{(where } \vec{a} = \hat{i} + \hat{j} + \hat{k} \text{)} \]

### Step 4: Substituting Assumed Values
Substituting these values into the equation:
\[ 3\vec{a} + 8\vec{a} = 6\vec{a} + 5\vec{a} \]
This simplifies to:
\[ 11\vec{a} = 11\vec{a} \]
This equation holds true, indicating that the vectors can indeed represent the same point.

### Step 5: Conclusion on the Relationship
Since we have shown that both lines PQ and RS can represent the same point when the vectors are equal, we conclude that the lines intersect.

### Final Answer
The lines PQ and RS are intersecting.

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