If `veca, vecb, vecc` be three units vectors perpendicular to each other, then `|(2veca+3vecb+4vecc).(veca xx vecb+5vecbxxvecc+6vecc xx veca)|`is equal to

Options

A. 18
B. 0
C. 4
D. 32

Answer

To solve the problem, we need to evaluate the expression:

\[ |(2\vec{a} + 3\vec{b} + 4\vec{c}) \cdot (\vec{a} \times \vec{b} + 5\vec{b} \times \vec{c} + 6\vec{c} \times \vec{a})| \]

where \(\vec{a}, \vec{b}, \vec{c}\) are unit vectors that are mutually perpendicular.

### Step 1: Understand the Cross Products
Since \(\vec{a}, \vec{b}, \vec{c}\) are unit vectors and mutually perpendicular, we can express the cross products:
- \(\vec{a} \times \vec{b} = \vec{c}\)
- \(\vec{b} \times \vec{c} = \vec{a}\)
- \(\vec{c} \times \vec{a} = \vec{b}\)

### Step 2: Substitute the Cross Products
Now substitute these results into the expression:

\[ \vec{a} \times \vec{b} + 5\vec{b} \times \vec{c} + 6\vec{c} \times \vec{a} = \vec{c} + 5\vec{a} + 6\vec{b} \]

### Step 3: Rewrite the Dot Product
Now we rewrite the dot product:

\[ (2\vec{a} + 3\vec{b} + 4\vec{c}) \cdot (\vec{c} + 5\vec{a} + 6\vec{b}) \]

### Step 4: Expand the Dot Product
Using the distributive property of the dot product, we expand it:

\[ = 2\vec{a} \cdot \vec{c} + 10\vec{a} \cdot \vec{a} + 12\vec{a} \cdot \vec{b} + 3\vec{b} \cdot \vec{c} + 15\vec{b} \cdot \vec{a} + 18\vec{b} \cdot \vec{b} + 4\vec{c} \cdot \vec{c} + 20\vec{c} \cdot \vec{a} + 24\vec{c} \cdot \vec{b} \]

### Step 5: Evaluate Each Dot Product
Since \(\vec{a}, \vec{b}, \vec{c}\) are unit vectors and perpendicular:

- \(\vec{a} \cdot \vec{a} = 1\)
- \(\vec{b} \cdot \vec{b} = 1\)
- \(\vec{c} \cdot \vec{c} = 1\)
- \(\vec{a} \cdot \vec{b} = 0\)
- \(\vec{b} \cdot \vec{c} = 0\)
- \(\vec{c} \cdot \vec{a} = 0\)

Thus, the expression simplifies to:

\[ = 10(1) + 18(1) + 4(1) = 10 + 18 + 4 = 32 \]

### Step 6: Absolute Value
Since we are interested in the absolute value:

\[ |(2\vec{a} + 3\vec{b} + 4\vec{c}) \cdot (\vec{c} + 5\vec{a} + 6\vec{b})| = |32| = 32 \]

### Final Answer
Thus, the final answer is:

\[ \boxed{32} \]