The value of `[[overline(a)-overline(b), overline(b)-overline(c), overline(c)-overline(a)]]` where` |overline(a)|=1, |overline(b)|=2 and |overline(c)|=3` is

Options

A. `1`
B. `6`
C. `0`
D. `3`

Answer

To solve the problem, we need to find the value of the scalar triple product given by:

\[ [[\overline{a} - \overline{b}, \overline{b} - \overline{c}, \overline{c} - \overline{a}]] \]

where \(|\overline{a}| = 1\), \(|\overline{b}| = 2\), and \(|\overline{c}| = 3\).

### Step 1: Rewrite the vectors
We can denote:
- \(\overline{a} = \mathbf{a}\)
- \(\overline{b} = 2\mathbf{b}\)
- \(\overline{c} = 3\mathbf{c}\)

Thus, we need to evaluate:

\[ [[\mathbf{a} - 2\mathbf{b}, 2\mathbf{b} - 3\mathbf{c}, 3\mathbf{c} - \mathbf{a}]] \]

### Step 2: Simplify the vectors
Now, we can simplify the vectors in the triple product:

1. The first vector is \(\mathbf{a} - 2\mathbf{b}\).
2. The second vector is \(2\mathbf{b} - 3\mathbf{c}\).
3. The third vector is \(3\mathbf{c} - \mathbf{a}\).

### Step 3: Apply the property of scalar triple product
Using the property of scalar triple product, we can perform column operations. We can add or subtract columns without changing the value of the scalar triple product.

Let's perform the following operation:

- Replace the first column with \((\mathbf{a} - 2\mathbf{b}) + (3\mathbf{c} - \mathbf{a})\).

This results in:

\[ (\mathbf{a} - 2\mathbf{b} + 3\mathbf{c} - \mathbf{a}) = (3\mathbf{c} - 2\mathbf{b}) \]

Now, the matrix becomes:

\[ [[3\mathbf{c} - 2\mathbf{b}, 2\mathbf{b} - 3\mathbf{c}, 3\mathbf{c} - \mathbf{a}]] \]

### Step 4: Observe the cancellation
Now, if we look closely at the columns, we can see that:

1. The first column is \(3\mathbf{c} - 2\mathbf{b}\).
2. The second column is \(2\mathbf{b} - 3\mathbf{c}\).
3. The third column is \(3\mathbf{c} - \mathbf{a}\).

Notice that if we add the first two columns, we have:

\[ (3\mathbf{c} - 2\mathbf{b}) + (2\mathbf{b} - 3\mathbf{c}) = 0 \]

### Step 5: Conclusion
Since one of the columns becomes zero, the scalar triple product is zero:

\[ [[\overline{a} - \overline{b}, \overline{b} - \overline{c}, \overline{c} - \overline{a}]] = 0 \]

Thus, the final answer is:

\[ \text{The value is } 0. \]