`veca , vecb , vecc` are mutually `bot` unit vectors equally inclined to `veca+vecb+vecc` at an angle `theta` then find the value of `36cos^2(2theta)`

Answer

To solve the problem, we need to find the value of \( 36 \cos^2(2\theta) \) given that the vectors \( \vec{a}, \vec{b}, \vec{c} \) are mutually perpendicular unit vectors that are equally inclined to \( \vec{a} + \vec{b} + \vec{c} \) at an angle \( \theta \).

### Step-by-Step Solution:

1. **Understanding the Vectors**:
Since \( \vec{a}, \vec{b}, \vec{c} \) are mutually perpendicular unit vectors, we can represent them as:
\[ \vec{a} = \hat{i}, \quad \vec{b} = \hat{j}, \quad \vec{c} = \hat{k} \]

2. **Finding the Sum of Vectors**:
The sum of these vectors is:
\[ \vec{a} + \vec{b} + \vec{c} = \hat{i} + \hat{j} + \hat{k} \]

3. **Calculating the Magnitude of the Sum**:
The magnitude of \( \vec{a} + \vec{b} + \vec{c} \) is:
\[ |\vec{a} + \vec{b} + \vec{c}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} \]

4. **Using the Dot Product**:
The angle \( \theta \) between \( \vec{a} \) and \( \vec{a} + \vec{b} + \vec{c} \) can be found using the dot product:
\[ \vec{a} \cdot (\vec{a} + \vec{b} + \vec{c}) = |\vec{a}| |\vec{a} + \vec{b} + \vec{c}| \cos \theta \]
Here, \( |\vec{a}| = 1 \) and \( |\vec{a} + \vec{b} + \vec{c}| = \sqrt{3} \):
\[ \vec{a} \cdot (\vec{a} + \vec{b} + \vec{c}) = \hat{i} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1 \]
Therefore,
\[ 1 = 1 \cdot \sqrt{3} \cos \theta \implies \cos \theta = \frac{1}{\sqrt{3}} \]

5. **Finding \( \cos(2\theta) \)**:
We use the double angle formula:
\[ \cos(2\theta) = 2\cos^2(\theta) - 1 \]
Substituting \( \cos \theta = \frac{1}{\sqrt{3}} \):
\[ \cos^2(\theta) = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \]
Thus,
\[ \cos(2\theta) = 2 \cdot \frac{1}{3} - 1 = \frac{2}{3} - 1 = -\frac{1}{3} \]

6. **Calculating \( \cos^2(2\theta) \)**:
Now we find \( \cos^2(2\theta) \):
\[ \cos^2(2\theta) = \left(-\frac{1}{3}\right)^2 = \frac{1}{9} \]

7. **Final Calculation**:
Finally, we compute \( 36 \cos^2(2\theta) \):
\[ 36 \cos^2(2\theta) = 36 \cdot \frac{1}{9} = 4 \]

### Conclusion:
The value of \( 36 \cos^2(2\theta) \) is \( \boxed{4} \).