If `veca,vecb,vecc` are linearly independent vectors, then `((veca+2vecb)xx(2vecb+vecc).(5vecc+veca))/(veca.(vecbxxvecc))` is equal to

Options

A. 10
B. 14
C. 18
D. 12

Explanation

To solve the given problem, we will follow a systematic approach to simplify the expression step by step.

**Given Expression:**
\[ \frac{(\vec{a} + 2\vec{b}) \times (2\vec{b} + \vec{c}) \cdot (5\vec{c} + \vec{a})}{\vec{a} \cdot (\vec{b} \times \vec{c})} \]

### Step 1: Expand the Cross Product
We start by expanding the cross product in the numerator:
\[ (\vec{a} + 2\vec{b}) \times (2\vec{b} + \vec{c}) = \vec{a} \times (2\vec{b} + \vec{c}) + 2\vec{b} \times (2\vec{b} + \vec{c}) \]

### Step 2: Simplify Each Term
Now we simplify each term:
1. **First Term:**
\[ \vec{a} \times (2\vec{b} + \vec{c}) = 2(\vec{a} \times \vec{b}) + \vec{a} \times \vec{c} \]

2. **Second Term:**
\[ 2\vec{b} \times (2\vec{b} + \vec{c}) = 2\vec{b} \times 2\vec{b} + 2\vec{b} \times \vec{c} = 0 + 2(\vec{b} \times \vec{c}) = 2(\vec{b} \times \vec{c}) \]

Combining these results, we have:
\[ (\vec{a} + 2\vec{b}) \times (2\vec{b} + \vec{c}) = 2(\vec{a} \times \vec{b}) + \vec{a} \times \vec{c} + 2(\vec{b} \times \vec{c}) \]

### Step 3: Dot Product with \( (5\vec{c} + \vec{a}) \)
Now we take the dot product of the result with \( (5\vec{c} + \vec{a}) \):
\[ (2(\vec{a} \times \vec{b}) + \vec{a} \times \vec{c} + 2(\vec{b} \times \vec{c})) \cdot (5\vec{c} + \vec{a}) \]

Distributing the dot product:
1. **First Term:**
\[ 2(\vec{a} \times \vec{b}) \cdot 5\vec{c} + 2(\vec{a} \times \vec{b}) \cdot \vec{a} = 10(\vec{a} \times \vec{b}) \cdot \vec{c} + 0 = 10(\vec{a} \times \vec{b}) \cdot \vec{c} \]

2. **Second Term:**
\[ (\vec{a} \times \vec{c}) \cdot 5\vec{c} + (\vec{a} \times \vec{c}) \cdot \vec{a} = 0 + 0 = 0 \]

3. **Third Term:**
\[ 2(\vec{b} \times \vec{c}) \cdot 5\vec{c} + 2(\vec{b} \times \vec{c}) \cdot \vec{a} = 0 + 2(\vec{b} \times \vec{c}) \cdot \vec{a} = 2(\vec{b} \times \vec{c}) \cdot \vec{a} \]

Combining these results gives:
\[ 10(\vec{a} \times \vec{b}) \cdot \vec{c} + 2(\vec{b} \times \vec{c}) \cdot \vec{a} \]

### Step 4: Final Expression
So, the numerator becomes:
\[ 10(\vec{a} \times \vec{b}) \cdot \vec{c} + 2(\vec{b} \times \vec{c}) \cdot \vec{a} \]

### Step 5: Divide by the Denominator
Now we divide by the denominator:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = (\vec{a} \times \vec{b} \times \vec{c}) \]

### Step 6: Simplifying the Expression
Using the scalar triple product properties, we can simplify:
\[ \frac{10(\vec{a} \times \vec{b}) \cdot \vec{c} + 2(\vec{b} \times \vec{c}) \cdot \vec{a}}{\vec{a} \cdot (\vec{b} \times \vec{c})} \]

Using the identity of scalar triple products, we find:
\[ = 10 + 2 = 12 \]

### Final Answer
Thus, the value of the given expression is:
\[ \boxed{12} \]