Step-by-step Solution:

1. Understanding the Geometry and Vectors

Let PQRSTU be a regular hexagon. The vectors $\"{PQ}\", \"{QR}\", \"{RS}\", \"{ST}\", \"{TU}\",$ and $\vec{UP}$ represent its sides taken in order. Let the side length of the hexagon be 'a'.

Key properties of a regular hexagon:

• All sides are equal in length, i.e., $|\vec{PQ}| = |\vec{QR}| = ... = a$.
• The interior angle between any two adjacent sides is 120°.
• Opposite sides are parallel and equal in magnitude. Therefore, we have:
  • $\vec{ST} = -\vec{PQ}$
  • $\vec{TU} = -\vec{QR}$
  • $\vec{UP} = -\vec{RS}$

The question contains a symbol 'e', which is likely a typo for '$\ne$' (not equal to). We will proceed with this standard interpretation.

2. Analyzing Statement 1

Statement 1: $\vec{PQ} \times (\vec{RS} + \vec{ST}) \ne \vec{0}$

First, let's simplify the term in the parenthesis using the triangle law of vector addition. For triangle RST, we have $\vec{RS} + \vec{ST} = \vec{RT}$. So, Statement 1 is equivalent to checking if $\vec{PQ} \times \vec{RT} \ne \vec{0}$.

The cross product of two non-zero vectors is zero if and only if they are parallel or anti-parallel. So, we need to determine if the vector $\vec{PQ}$ is parallel to the vector $\vec{RT}$.

• $\vec{PQ}$ is a side of the hexagon.
• $\vec{RT}$ is a diagonal of the hexagon connecting vertices R and T.

Geometrically, in a regular hexagon, a side vector is not parallel to any diagonal except the main diagonals (like PS, QT, RU), and RT is not a main diagonal. Let's verify this more rigorously.

Let's place vertex P at the origin (0,0) and Q on the x-axis at (a, 0). Then $\vec{PQ} = a\hat{i}$.

The vertices can be found by rotating subsequent side vectors by 60° (exterior angle).

• $\vec{QR} = a(\cos 60^{\circ} \hat{i} + \sin 60^{\circ} \hat{j}) = a(\frac{1}{2}\hat{i} + \frac{\sqrt{3}}{2}\hat{j})$
• $\vec{RS} = a(\cos 120^{\circ} \hat{i} + \sin 120^{\circ} \hat{j}) = a(-\frac{1}{2}\hat{i} + \frac{\sqrt{3}}{2}\hat{j})$
• $\vec{ST} = a(\cos 180^{\circ} \hat{i} + \sin 180^{\circ} \hat{j}) = -a\hat{i}$

Now, we find the vector $\vec{RT}$. $\vec{RT} = \vec{RS} + \vec{ST} = a(-\frac{1}{2}\hat{i} + \frac{\sqrt{3}}{2}\hat{j}) + (-a\hat{i}) = a(-\frac{3}{2}\hat{i} + \frac{\sqrt{3}}{2}\hat{j})$

The vector $\vec{PQ}$ is purely in the $\hat{i}$ direction, while the vector $\vec{RT}$ has both $\hat{i}$ and $\hat{j}$ components. Thus, $\vec{PQ}$ and $\vec{RT}$ are not parallel.

Therefore, their cross product is non-zero: $\vec{PQ} \times \vec{RT} = (a\hat{i}) \times a(-\frac{3}{2}\hat{i} + \frac{\sqrt{3}}{2}\hat{j}) = a^2 (\hat{i} \times (-\frac{3}{2}\hat{i}) + \hat{i} \times (\frac{\sqrt{3}}{2}\hat{j})) = a^2 (\vec{0} + \frac{\sqrt{3}}{2}\hat{k}) = \frac{a^2\sqrt{3}}{2}\hat{k} \ne \vec{0}$

Thus, Statement 1 is True.

3. Analyzing Statement 2

Statement 2: $\vec{PQ} \times \vec{RS} = \vec{0}$ and $\vec{PQ} \times \vec{ST} \ne \vec{0}$

This statement is a conjunction of two claims. For the entire statement to be true, both claims must be true.

• Claim A: $\vec{PQ} \times \vec{RS} = \vec{0}$ This would mean $\vec{PQ}$ and $\vec{RS}$ are parallel. However, $\vec{PQ}$ and $\vec{RS}$ are sides of the regular hexagon separated by the side $\vec{QR}$. The angle between them is 120°. Since the angle is not 0° or 180°, the vectors are not parallel. Therefore, their cross product is not zero. $\vec{PQ} \times \vec{RS} \ne \vec{0}$. So, Claim A is False.

Since one part of the conjunction is false, the entire Statement 2 is false. For completeness, let's also check the second claim.

• Claim B: $\vec{PQ} \times \vec{ST} \ne \vec{0}$ This would mean $\vec{PQ}$ and $\vec{ST}$ are not parallel. However, in a regular hexagon, opposite sides are parallel. $\vec{PQ}$ and $\vec{ST}$ are opposite sides. Specifically, they are anti-parallel, so $\vec{ST} = -\vec{PQ}$. The cross product of parallel or anti-parallel vectors is the zero vector. $\vec{PQ} \times \vec{ST} = \vec{PQ} \times (-\vec{PQ}) = -(\vec{PQ} \times \vec{PQ}) = -\vec{0} = \vec{0}$. So, the claim that $\vec{PQ} \times \vec{ST} \ne \vec{0}$ is False.

Since both claims are false, Statement 2 is definitively False.

4. Conclusion

• Statement 1 is True.
• Statement 2 is False.

This corresponds to option C.

Final Answer: The final answer is $\boxed{C}$