Let the vectors PQ,OR,RS,ST,TU and UP represent the sides of a regular hexagon.
Statement I: `PQxx(RS+ST)ne0`
Statement II: `PQxxRS=0 and PQxxSTne0`

Options

A. Both statement I and statement II are correct and statement II is the correct explanation of statement I
B. both statement I and statement II are correct but statement II is not the correct explanation of statement I
C. Statement I is correct but statement II is incorrect
D. Statement II is correct but statement I is incorrect

Answer

To solve the problem, we need to analyze the two statements given regarding the vectors representing the sides of a regular hexagon.

### Step-by-Step Solution:

1. **Understanding the Geometry of a Regular Hexagon**:
- A regular hexagon has equal sides and angles. Let's denote the vertices of the hexagon as \( P, Q, R, S, T, U \).
- The vectors representing the sides of the hexagon are \( \vec{PQ}, \vec{QR}, \vec{RS}, \vec{ST}, \vec{TU}, \vec{UP} \).

2. **Analyzing Statement I**:
- Statement I states: \( \vec{PQ} \times (\vec{RS} + \vec{ST}) \neq 0 \).
- We can rewrite this using the distributive property of the cross product:
\[ \vec{PQ} \times \vec{RS} + \vec{PQ} \times \vec{ST} \neq 0 \]

3. **Understanding the Cross Products**:
- Since \( \vec{PQ} \) and \( \vec{ST} \) are parallel (as they are opposite sides of the hexagon), we have:
\[ \vec{PQ} \times \vec{ST} = 0 \]
- However, \( \vec{PQ} \) and \( \vec{RS} \) are not parallel; they form an angle \( \theta \) (where \( 0 < \theta < 180^\circ \)). Thus:
\[ \vec{PQ} \times \vec{RS} \neq 0 \]

4. **Conclusion for Statement I**:
- Since \( \vec{PQ} \times \vec{RS} \) is a non-zero vector and \( \vec{PQ} \times \vec{ST} = 0 \), we can conclude:
\[ \vec{PQ} \times (\vec{RS} + \vec{ST}) = \vec{PQ} \times \vec{RS} + 0 = \vec{PQ} \times \vec{RS} \neq 0 \]
- Therefore, Statement I is **True**.

5. **Analyzing Statement II**:
- Statement II states: \( \vec{PQ} \times \vec{RS} = 0 \) and \( \vec{PQ} \times \vec{ST} \neq 0 \).
- From our previous analysis, we know:
- \( \vec{PQ} \times \vec{RS} \neq 0 \) (this is false).
- \( \vec{PQ} \times \vec{ST} = 0 \) (this is true).

6. **Conclusion for Statement II**:
- Since one part of Statement II is false, we conclude that Statement II is **False**.

### Final Answer:
- Statement I is True, and Statement II is False.