Let `veca,` `vecb` and `vecc` be three vectors such that `|veca |=sqrt(3),` `|vec b|=5,` `vec b .vec c = 10` and the angle between `vec b` and `vec c` is `pi/3,` if `vec a` is perpendicular to the vector `vec b xx vec c,` then `|veca xx (vecbxxvecc)|` is equal to __________.

Answer

To solve the problem, we need to find the magnitude of the vector \( \vec{a} \times (\vec{b} \times \vec{c}) \) given the conditions provided.

### Step 1: Find the magnitude of \( \vec{c} \)

We know that:
\[ \vec{b} \cdot \vec{c} = |\vec{b}| |\vec{c}| \cos(\theta) \]
where \( \theta \) is the angle between \( \vec{b} \) and \( \vec{c} \). Given:
- \( |\vec{b}| = 5 \)
- \( \vec{b} \cdot \vec{c} = 10 \)
- \( \theta = \frac{\pi}{3} \)

Substituting the values:
\[ 10 = 5 |\vec{c}| \cos\left(\frac{\pi}{3}\right) \]
Since \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \):
\[ 10 = 5 |\vec{c}| \cdot \frac{1}{2} \]
\[ 10 = \frac{5}{2} |\vec{c}| \]
Multiplying both sides by 2:
\[ 20 = 5 |\vec{c}| \]
Dividing by 5:
\[ |\vec{c}| = 4 \]

### Step 2: Calculate \( |\vec{b} \times \vec{c}| \)

The magnitude of the cross product \( |\vec{b} \times \vec{c}| \) is given by:
\[ |\vec{b} \times \vec{c}| = |\vec{b}| |\vec{c}| \sin(\theta) \]
Substituting the known values:
\[ |\vec{b} \times \vec{c}| = 5 \cdot 4 \cdot \sin\left(\frac{\pi}{3}\right) \]
Since \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \):
\[ |\vec{b} \times \vec{c}| = 5 \cdot 4 \cdot \frac{\sqrt{3}}{2} = 10\sqrt{3} \]

### Step 3: Calculate \( |\vec{a} \times (\vec{b} \times \vec{c})| \)

Since \( \vec{a} \) is perpendicular to \( \vec{b} \times \vec{c} \), we have:
\[ |\vec{a} \times (\vec{b} \times \vec{c})| = |\vec{a}| |\vec{b} \times \vec{c}| \sin(90^\circ) \]
Given \( |\vec{a}| = \sqrt{3} \) and \( \sin(90^\circ) = 1 \):
\[ |\vec{a} \times (\vec{b} \times \vec{c})| = \sqrt{3} \cdot (10\sqrt{3}) \cdot 1 = 30 \]

### Final Answer
Thus, the magnitude \( |\vec{a} \times (\vec{b} \times \vec{c})| \) is equal to **30**.