P `(vecp) and Q (vecq)` are the position vectors of two fixed points and `R(vecr)` is the postion vector of a variable point. If R moves such that `(vecr-vecp)xx (vecr-vecq)=vec0`then the locus of R is

Options

A. a plane containing the origian O and parallel to two non-collinear vectors `vec(OP) and vec(OQ) `
B. the surface of a sphere described on PQ as its diameter
C. a line passing through points P and Q
D. a set of lines parallel to line PQ

Explanation

To solve the problem, we need to analyze the given condition involving the position vectors of points P, Q, and R. Let's go through the steps systematically.

### Step-by-Step Solution

1. **Understand the Given Condition**:
We are given that the position vectors of points P and Q are denoted by \(\vec{p}\) and \(\vec{q}\), respectively. The position vector of the variable point R is denoted by \(\vec{r}\). The condition given is:
\[ (\vec{r} - \vec{p}) \times (\vec{r} - \vec{q}) = \vec{0} \]

2. **Interpret the Cross Product**:
The cross product of two vectors is zero if and only if the vectors are collinear. Therefore, the condition implies that the vectors \(\vec{r} - \vec{p}\) and \(\vec{r} - \vec{q}\) are collinear.

3. **Express Collinearity**:
If the vectors are collinear, we can express this relationship as:
\[ \vec{r} - \vec{p} = k(\vec{r} - \vec{q}) \]
for some scalar \(k\).

4. **Rearranging the Equation**:
Rearranging the equation gives:
\[ \vec{r} - \vec{p} = k\vec{r} - k\vec{q} \]
This can be rearranged to isolate \(\vec{r}\):
\[ \vec{r} - k\vec{r} = k\vec{q} + \vec{p} \]
\[ \vec{r}(1 - k) = k\vec{q} + \vec{p} \]
Hence,
\[ \vec{r} = \frac{k\vec{q} + \vec{p}}{1 - k} \]

5. **Interpret the Result**:
The expression shows that \(\vec{r}\) can be expressed as a linear combination of \(\vec{p}\) and \(\vec{q}\). This means that point R lies on the line that passes through points P and Q.

6. **Conclusion**:
Therefore, the locus of the point R is a straight line passing through the fixed points P and Q.

### Final Answer:
The locus of R is a straight line passing through points P and Q.