Statement 1: Let `theta` be the angle between the line `(x-2)/2=(y-1)/(-3)=(z+2)/(-2)` and the plane `x+y-z=5.` Then `theta=sin^(-1)(1//sqrt(51))dot` Statement 2: The angle between a straight line and a plane is the complement of the angle between the line and the normal to the plane.

Options

A. Both the statements are true, and Statement 2 is the correct explanation for Statement 1.
B. Both the Statements are true, but Statement 2 is not the correct explanation for Statement 1.
C. Statement 1 is true and Statement 2 is false.
D. Statement 1 is false and Statement 2 is true.

Explanation

To solve the problem, we need to find the angle θ between the given line and the plane. We will follow these steps:

### Step 1: Identify the Direction Vector of the Line
The line is given in the symmetric form:
\[ \frac{x-2}{2} = \frac{y-1}{-3} = \frac{z+2}{-2} \]
From this, we can identify the direction vector \( \mathbf{d} \) of the line as:
\[ \mathbf{d} = \langle 2, -3, -2 \rangle \]

### Step 2: Identify the Normal Vector of the Plane
The equation of the plane is given as:
\[ x + y - z = 5 \]
From this equation, we can identify the normal vector \( \mathbf{n} \) of the plane as:
\[ \mathbf{n} = \langle 1, 1, -1 \rangle \]

### Step 3: Use the Dot Product to Find the Angle
The angle between the line and the normal vector of the plane can be found using the dot product formula:
\[ \mathbf{d} \cdot \mathbf{n} = |\mathbf{d}| |\mathbf{n}| \cos(90^\circ - \theta) \]
This simplifies to:
\[ \mathbf{d} \cdot \mathbf{n} = |\mathbf{d}| |\mathbf{n}| \sin(\theta) \]

### Step 4: Calculate the Dot Product
Calculating the dot product \( \mathbf{d} \cdot \mathbf{n} \):
\[ \mathbf{d} \cdot \mathbf{n} = 2 \cdot 1 + (-3) \cdot 1 + (-2) \cdot (-1) = 2 - 3 + 2 = 1 \]

### Step 5: Calculate the Magnitudes of the Vectors
Now, we calculate the magnitudes of \( \mathbf{d} \) and \( \mathbf{n} \):
\[ |\mathbf{d}| = \sqrt{2^2 + (-3)^2 + (-2)^2} = \sqrt{4 + 9 + 4} = \sqrt{17} \]
\[ |\mathbf{n}| = \sqrt{1^2 + 1^2 + (-1)^2} = \sqrt{1 + 1 + 1} = \sqrt{3} \]

### Step 6: Substitute into the Equation
Substituting the values into the equation:
\[ 1 = \sqrt{17} \cdot \sqrt{3} \cdot \sin(\theta) \]
This simplifies to:
\[ \sin(\theta) = \frac{1}{\sqrt{51}} \]

### Step 7: Find the Angle θ
Finally, we can find θ:
\[ \theta = \sin^{-1}\left(\frac{1}{\sqrt{51}}\right) \]

### Conclusion
Thus, we have shown that:
\[ \text{Statement 1 is true: } \theta = \sin^{-1}\left(\frac{1}{\sqrt{51}}\right) \]
And we have verified that:
\[ \text{Statement 2 is true: The angle between a straight line and a plane is the complement of the angle between the line and the normal to the plane.} \]