Let `|veca| = 3, |vecb| = 5, vecb.vecc= 10`, angle between `vecb` and `vecc` equal to `pi/3`. If `veca` is perpendicular `vecb xx vecc` then find the value of `|veca xx (vecb xx vecc)| `

Answer

To solve the problem step by step, let's break it down:

### Given:
- \(|\vec{a}| = 3\)
- \(|\vec{b}| = 5\)
- \(\vec{b} \cdot \vec{c} = 10\)
- Angle between \(\vec{b}\) and \(\vec{c} = \frac{\pi}{3}\)
- \(\vec{a}\) is perpendicular to \(\vec{b} \times \vec{c}\)

### To Find:
- \(|\vec{a} \times (\vec{b} \times \vec{c})|\)

### Step 1: Calculate \(|\vec{c}|\)
Using the dot product formula:
\[ \vec{b} \cdot \vec{c} = |\vec{b}| |\vec{c}| \cos(\theta) \]
where \(\theta\) is the angle between \(\vec{b}\) and \(\vec{c}\).

Substituting the known values:
\[ 10 = 5 |\vec{c}| \cos\left(\frac{\pi}{3}\right) \]
Since \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\), we have:
\[ 10 = 5 |\vec{c}| \cdot \frac{1}{2} \]
\[ 10 = \frac{5}{2} |\vec{c}| \]
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 \(|\vec{b} \times \vec{c}|\) can be calculated using:
\[ |\vec{b} \times \vec{c}| = |\vec{b}| |\vec{c}| \sin(\theta) \]
where \(\theta\) is the angle between \(\vec{b}\) and \(\vec{c}\).

Substituting the known values:
\[ |\vec{b} \times \vec{c}| = 5 \cdot 4 \cdot \sin\left(\frac{\pi}{3}\right) \]
Since \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\):
\[ |\vec{b} \times \vec{c}| = 5 \cdot 4 \cdot \frac{\sqrt{3}}{2} \]
\[ |\vec{b} \times \vec{c}| = 10\sqrt{3} \]

### Step 3: Calculate \(|\vec{a} \times (\vec{b} \times \vec{c})|\)
Using the vector triple product identity:
\[ |\vec{a} \times (\vec{b} \times \vec{c})| = |\vec{a}| |\vec{b} \times \vec{c}| \sin(\phi) \]
where \(\phi\) is the angle between \(\vec{a}\) and \(\vec{b} \times \vec{c}\). Since \(\vec{a}\) is perpendicular to \(\vec{b} \times \vec{c}\), \(\phi = \frac{\pi}{2}\) and \(\sin\left(\frac{\pi}{2}\right) = 1\).

Substituting the known values:
\[ |\vec{a} \times (\vec{b} \times \vec{c})| = |\vec{a}| |\vec{b} \times \vec{c}| \]
\[ |\vec{a} \times (\vec{b} \times \vec{c})| = 3 \cdot 10\sqrt{3} \]
\[ |\vec{a} \times (\vec{b} \times \vec{c})| = 30\sqrt{3} \]

### Final Answer:
\[ |\vec{a} \times (\vec{b} \times \vec{c})| = 30\sqrt{3} \]