If the angle between the plane `x-3y+2z=1` and the line `(x-1)/2=(y-1)/1=(z-1)/(-3)` is `theta,` then the find the value of `cosec theta`.

Explanation

To solve the problem, we need to find the value of \( \csc \theta \) where \( \theta \) is the angle between the plane \( x - 3y + 2z = 1 \) and the line given by the equations \( \frac{x-1}{2} = \frac{y-1}{1} = \frac{z-1}{-3} \).

### Step 1: Identify the normal vector of the plane
The equation of the plane is given as:
\[ x - 3y + 2z = 1 \]
From this equation, we can identify the normal vector \( \mathbf{n} \) of the plane:
\[ \mathbf{n} = \langle 1, -3, 2 \rangle \]

### Step 2: Identify the direction vector of the line
The line is given in symmetric form:
\[ \frac{x-1}{2} = \frac{y-1}{1} = \frac{z-1}{-3} \]
From this, we can extract the direction ratios of the line, which gives us the direction vector \( \mathbf{a} \):
\[ \mathbf{a} = \langle 2, 1, -3 \rangle \]

### Step 3: Find the angle between the normal vector and the direction vector
Let \( \alpha \) be the angle between the normal vector \( \mathbf{n} \) and the direction vector \( \mathbf{a} \). The relationship between \( \alpha \) and \( \theta \) is:
\[ \alpha + \theta = 90^\circ \quad \Rightarrow \quad \alpha = 90^\circ - \theta \]
Thus, we can use the cosine of \( \alpha \):
\[ \cos \alpha = \frac{\mathbf{n} \cdot \mathbf{a}}{|\mathbf{n}| |\mathbf{a}|} \]

### Step 4: Calculate the dot product \( \mathbf{n} \cdot \mathbf{a} \)
Calculating the dot product:
\[ \mathbf{n} \cdot \mathbf{a} = (1)(2) + (-3)(1) + (2)(-3) = 2 - 3 - 6 = -7 \]

### Step 5: Calculate the magnitudes of \( \mathbf{n} \) and \( \mathbf{a} \)
Calculating the magnitude of \( \mathbf{n} \):
\[ |\mathbf{n}| = \sqrt{1^2 + (-3)^2 + 2^2} = \sqrt{1 + 9 + 4} = \sqrt{14} \]

Calculating the magnitude of \( \mathbf{a} \):
\[ |\mathbf{a}| = \sqrt{2^2 + 1^2 + (-3)^2} = \sqrt{4 + 1 + 9} = \sqrt{14} \]

### Step 6: Substitute into the cosine formula
Now substituting into the cosine formula:
\[ \cos \alpha = \frac{-7}{\sqrt{14} \cdot \sqrt{14}} = \frac{-7}{14} = -\frac{1}{2} \]

### Step 7: Relate \( \cos \alpha \) to \( \sin \theta \)
Since \( \alpha = 90^\circ - \theta \):
\[ \cos(90^\circ - \theta) = \sin \theta \]
Thus,
\[ \sin \theta = -\frac{1}{2} \]
However, since sine cannot be negative for angles in the range of \( 0^\circ \) to \( 90^\circ \), we take the positive value:
\[ \sin \theta = \frac{1}{2} \]

### Step 8: Find \( \csc \theta \)
The cosecant is the reciprocal of sine:
\[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{1}{2}} = 2 \]

### Final Answer
Thus, the value of \( \csc \theta \) is:
\[ \boxed{2} \]