For three non - zero vectors `veca, vecb and vecc`, if `[(veca, vecb , vecc)]=4`, then `[(vecaxx(vecb+2vecc), vecbxx(vecc-3veca), vecc xx(3veca+vecb))]`is equal to

Options

A. 12
B. 16
C. 84
D. 144

Answer

To solve the problem, we need to evaluate the expression given the scalar triple product of three non-zero vectors \(\vec{a}, \vec{b}, \vec{c}\) is equal to 4. The expression we need to evaluate is:

\[ [(\vec{a} \times (\vec{b} + 2\vec{c}), \vec{b} \times (\vec{c} - 3\vec{a}), \vec{c} \times (3\vec{a} + \vec{b}))] \]

### Step 1: Expand the Cross Products

First, we will expand each of the cross products in the expression:

1. \(\vec{a} \times (\vec{b} + 2\vec{c}) = \vec{a} \times \vec{b} + 2\vec{a} \times \vec{c}\)
2. \(\vec{b} \times (\vec{c} - 3\vec{a}) = \vec{b} \times \vec{c} - 3\vec{b} \times \vec{a}\)
3. \(\vec{c} \times (3\vec{a} + \vec{b}) = 3\vec{c} \times \vec{a} + \vec{c} \times \vec{b}\)

### Step 2: Substitute Back into the Expression

Now we substitute these expansions back into the expression:

\[ [(\vec{a} \times \vec{b} + 2\vec{a} \times \vec{c}), (\vec{b} \times \vec{c} - 3\vec{b} \times \vec{a}), (3\vec{c} \times \vec{a} + \vec{c} \times \vec{b})] \]

### Step 3: Rearranging the Terms

Now we can rearrange the terms to group similar vectors:

\[ [(\vec{a} \times \vec{b}), (\vec{b} \times \vec{c}), (3\vec{c} \times \vec{a})] + 2[(\vec{a} \times \vec{c}), (-3\vec{b} \times \vec{a}), (\vec{c} \times \vec{b})] \]

### Step 4: Use Properties of Determinants

Using the properties of determinants and the scalar triple product, we can express this determinant as:

\[ = [\vec{a}, \vec{b}, \vec{c}] + 2 \cdot [\vec{a}, \vec{c}, \vec{b}] + 3 \cdot [\vec{c}, \vec{a}, \vec{b}] \]

### Step 5: Simplifying the Expression

We know that the scalar triple product is invariant under cyclic permutations, so we can write:

\[ = [\vec{a}, \vec{b}, \vec{c}] + 2 \cdot [\vec{a}, \vec{b}, \vec{c}] + 3 \cdot [\vec{a}, \vec{b}, \vec{c}] \]

This simplifies to:

\[ = (1 + 2 + 3) \cdot [\vec{a}, \vec{b}, \vec{c}] = 6 \cdot [\vec{a}, \vec{b}, \vec{c}] \]

### Step 6: Substitute the Given Value

Given that \([\vec{a}, \vec{b}, \vec{c}] = 4\), we substitute this value in:

\[ = 6 \cdot 4 = 24 \]

### Final Result

Thus, the value of the expression is:

\[ \boxed{24} \]