If a unit vector`vec(a)` makes angle `(pi)/(3)` with `hat(i), (pi)/(4)` with `hat(j)` and `theta in ((pi)/(2), pi)` with `hat(k)` then a value of `theta` is `(k pi)/(l)` (k and l are coprime). The value of k is _________.

Explanation

To solve the problem, we need to find the value of \( \theta \) for the unit vector \( \vec{a} \) that makes specific angles with the coordinate axes. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the angles with the axes
Given that the unit vector \( \vec{a} \) makes an angle of \( \frac{\pi}{3} \) with \( \hat{i} \), \( \frac{\pi}{4} \) with \( \hat{j} \), and \( \theta \) with \( \hat{k} \), we can express the components of the vector in terms of these angles.

### Step 2: Write the components of the unit vector
The components of the unit vector \( \vec{a} \) can be expressed using the cosine of the angles:
- \( a_x = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \)
- \( a_y = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \)
- \( a_z = \cos(\theta) \)

### Step 3: Use the property of unit vectors
Since \( \vec{a} \) is a unit vector, it must satisfy the equation:
\[ a_x^2 + a_y^2 + a_z^2 = 1 \]
Substituting the values we found:
\[ \left(\frac{1}{2}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 + \cos^2(\theta) = 1 \]

### Step 4: Simplify the equation
Calculating the squares:
\[ \frac{1}{4} + \frac{1}{2} + \cos^2(\theta) = 1 \]
Converting \( \frac{1}{2} \) to a fraction with a common denominator:
\[ \frac{1}{4} + \frac{2}{4} + \cos^2(\theta) = 1 \]
This simplifies to:
\[ \frac{3}{4} + \cos^2(\theta) = 1 \]

### Step 5: Solve for \( \cos^2(\theta) \)
Rearranging gives:
\[ \cos^2(\theta) = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 6: Find \( \cos(\theta) \)
Taking the square root:
\[ \cos(\theta) = \pm \frac{1}{2} \]

### Step 7: Determine the possible values of \( \theta \)
The angles corresponding to \( \cos(\theta) = \frac{1}{2} \) and \( \cos(\theta) = -\frac{1}{2} \) are:
- \( \theta = \frac{\pi}{3} \) (not in the range \( \left(\frac{\pi}{2}, \pi\right) \))
- \( \theta = \frac{5\pi}{3} \) (not in the range \( \left(\frac{\pi}{2}, \pi\right) \))
- \( \theta = \frac{2\pi}{3} \) (in the range \( \left(\frac{\pi}{2}, \pi\right) \))

### Step 8: Express \( \theta \) in the required form
We can express \( \theta = \frac{2\pi}{3} \) in the form \( \frac{k\pi}{l} \) where \( k = 2 \) and \( l = 3 \).

### Conclusion
Since \( k \) and \( l \) are coprime, the value of \( k \) is:
\[ \boxed{2} \]