Let `veca` and `vecc` be unit vectors inclined at `pi//3` with each other. If `(veca xx (vecb xx vecc)). (veca xx vecc)=5`, then `[vecavecbvecc]` is equal to

Answer

To solve the problem step by step, we will analyze the given information and apply the properties of vectors and the scalar triple product.

### Step 1: Identify the Given Information
We have two unit vectors, \(\vec{a}\) and \(\vec{c}\), that are inclined at an angle of \(\frac{\pi}{3}\) radians (or 60 degrees) to each other. We also know that:
\[ (\vec{a} \times (\vec{b} \times \vec{c})) \cdot (\vec{a} \times \vec{c}) = 5 \]

### Step 2: Use the Scalar Triple Product Identity
We can use the identity for the scalar triple product:
\[ \vec{u} \times (\vec{v} \times \vec{w}) = (\vec{u} \cdot \vec{w}) \vec{v} - (\vec{u} \cdot \vec{v}) \vec{w} \]
Let \(\vec{u} = \vec{a}\), \(\vec{v} = \vec{b}\), and \(\vec{w} = \vec{c}\). Therefore:
\[ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]

### Step 3: Substitute into the Given Equation
Substituting this into the equation, we have:
\[ ((\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}) \cdot (\vec{a} \times \vec{c}) = 5 \]

### Step 4: Calculate \(\vec{a} \cdot \vec{c}\)
Since \(\vec{a}\) and \(\vec{c}\) are unit vectors and the angle between them is \(\frac{\pi}{3}\):
\[ \vec{a} \cdot \vec{c} = |\vec{a}| |\vec{c}| \cos\left(\frac{\pi}{3}\right) = 1 \cdot 1 \cdot \frac{1}{2} = \frac{1}{2} \]

### Step 5: Substitute \(\vec{a} \cdot \vec{c}\) into the Equation
Now substituting \(\vec{a} \cdot \vec{c} = \frac{1}{2}\):
\[ \left(\frac{1}{2} \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}\right) \cdot (\vec{a} \times \vec{c}) = 5 \]

### Step 6: Expand the Dot Product
Expanding the dot product:
\[ \frac{1}{2} (\vec{b} \cdot (\vec{a} \times \vec{c})) - (\vec{a} \cdot \vec{b}) (\vec{c} \cdot (\vec{a} \times \vec{c})) = 5 \]
Since \(\vec{c} \cdot (\vec{a} \times \vec{c}) = 0\) (as \(\vec{c}\) is parallel to itself), we simplify to:
\[ \frac{1}{2} (\vec{b} \cdot (\vec{a} \times \vec{c})) = 5 \]

### Step 7: Solve for \(\vec{b} \cdot (\vec{a} \times \vec{c})\)
Multiplying both sides by 2:
\[ \vec{b} \cdot (\vec{a} \times \vec{c}) = 10 \]

### Step 8: Relate to Volume of Parallelepiped
The expression \(\vec{b} \cdot (\vec{a} \times \vec{c})\) represents the volume of the parallelepiped formed by vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\). Thus, the volume \(V\) is:
\[ V = 10 \]

### Step 9: Find \([\vec{a} \vec{b} \vec{c}]\)
The volume of the parallelepiped can also be expressed as:
\[ [\vec{a} \vec{b} \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) \]
Since the order of the vectors matters, we have:
\[ [\vec{a} \vec{b} \vec{c}] = -10 \]

### Final Answer
Thus, the value of \([\vec{a} \vec{b} \vec{c}]\) is:
\[ \boxed{-10} \]