To solve the problem, we need to analyze each statement one by one and determine their validity.
### Step 1: Analyze Statement 1
**Statement 1:** The equation of the plane through points \( (2, 3, 3) \) and \( (1, -3, -4) \) and parallel to the line given by \( \frac{x-1}{3} = \frac{y-3}{4} = \frac{z+1}{5} \) is \( x + y - 7z = 16 \).
**Solution:**
1. **Find the direction ratios of the line:** The direction ratios from the line equation are \( (3, 4, 5) \).
2. **Find the vector from point A to point B:**
\[
\vec{AB} = (1 - 2, -3 - 3, -4 - 3) = (-1, -6, -7)
\]
3. **Use the normal vector:** The normal vector of the plane can be found using the cross product of \( \vec{AB} \) and the direction ratios of the line:
\[
\vec{n} = \vec{AB} \times (3, 4, 5)
\]
\[
\vec{n} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
-1 & -6 & -7 \\
3 & 4 & 5
\end{vmatrix}
\]
Calculating the determinant:
\[
\vec{n} = \hat{i}((-6)(5) - (-7)(4)) - \hat{j}((-1)(5) - (-7)(3)) + \hat{k}((-1)(4) - (-6)(3))
\]
\[
= \hat{i}(-30 + 28) - \hat{j}(-5 + 21) + \hat{k}(-4 + 18)
\]
\[
= \hat{i}(-2) - \hat{j}(16) + \hat{k}(14)
\]
Thus, the normal vector \( \vec{n} = (-2, -16, 14) \).
4. **Equation of the plane:** The equation of the plane can be written as:
\[
-2(x - 2) - 16(y - 3) + 14(z - 3) = 0
\]
Expanding this:
\[
-2x + 4 - 16y + 48 + 14z - 42 = 0
\]
\[
-2x - 16y + 14z + 10 = 0
\]
Dividing by -2 gives:
\[
x + 8y - 7z = -5
\]
This does not match the given equation \( x + y - 7z = 16 \). Therefore, **Statement 1 is false.**
### Step 2: Analyze Statement 2
**Statement 2:** The shortest distance between two non-intersecting lines \( \vec{r} = \vec{a} + \lambda \vec{b} \) and \( \vec{r} = \vec{c} + \mu \vec{d} \) is given by \( \frac{|\left[(\vec{c} - \vec{a}) \cdot (\vec{b} \times \vec{d})\right]|}{|\vec{b} \times \vec{d}|} \).
**Solution:**
The formula for the shortest distance between two skew lines is indeed given by:
\[
d = \frac{|(\vec{c} - \vec{a}) \cdot (\vec{b} \times \vec{d})|}{|\vec{b} \times \vec{d}|}
\]
Thus, **Statement 2 is true.**
### Step 3: Analyze Statement 3
**Statement 3:** The vector equation of a plane through a point having position vector \( \vec{a} \) and parallel to vectors \( \vec{b} \) and \( \vec{c} \) is \( \vec{r} = \vec{a} + \lambda \vec{b} + \mu \vec{c} \).
**Solution:**
This statement correctly describes the vector equation of a plane. Therefore, **Statement 3 is true.**
### Conclusion
- **Statement 1:** False
- **Statement 2:** True
- **Statement 3:** True
### Final Answer
The correct statements are 2 and 3, while statement 1 is false.
