If `vecu, vecv, vecw` are three vectors such that `[vecu vecv vecw]=1`, then
`3[vecu vecv vecw]-[vecv vecw vecu]-2[vecw vecv vecu]=`

Options

A. 0
B. 2
C. 3
D. 4

Explanation

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

\[ 3[\vec{u} \, \vec{v} \, \vec{w}] - [\vec{v} \, \vec{w} \, \vec{u}] - 2[\vec{w} \, \vec{v} \, \vec{u}] \]

Given that \([\vec{u} \, \vec{v} \, \vec{w}] = 1\), we can substitute this value into our expression.

### Step-by-step Solution:

1. **Identify the determinants**:
We know that \([\vec{u} \, \vec{v} \, \vec{w}] = 1\).

2. **Rearranging the determinants**:
We will rewrite the determinants in terms of \([\vec{u} \, \vec{v} \, \vec{w}]\):
- For \([\vec{v} \, \vec{w} \, \vec{u}]\), we can rearrange it:
\[ [\vec{v} \, \vec{w} \, \vec{u}] = -[\vec{u} \, \vec{v} \, \vec{w}] = -1 \]
- For \([\vec{w} \, \vec{v} \, \vec{u}]\), we can rearrange it as well:
\[ [\vec{w} \, \vec{v} \, \vec{u}] = -[\vec{u} \, \vec{w} \, \vec{v}] = -[\vec{u} \, \vec{v} \, \vec{w}] = -1 \]

3. **Substituting the values**:
Now substitute these values back into the expression:
\[ 3[\vec{u} \, \vec{v} \, \vec{w}] - [\vec{v} \, \vec{w} \, \vec{u}] - 2[\vec{w} \, \vec{v} \, \vec{u}] = 3(1) - (-1) - 2(-1) \]

4. **Calculating the expression**:
\[ = 3 + 1 + 2 = 6 \]

5. **Final Result**:
The final result of the expression is:
\[ 6 \]