`[[overline(a), overline(b), overline(a)timesoverline(b)]]=`

Options

A. `|overline(a)timesoverline(b)|`
B. `|overline(a)timesoverline(b)|^(2)`
C. `0`
D. `1`

Answer

To find the value of the scalar triple product \([ \overline{a}, \overline{b}, \overline{a} \times \overline{b} ]\), we can follow these steps:

### Step 1: Understand the Scalar Triple Product
The scalar triple product of three vectors \(\overline{a}\), \(\overline{b}\), and \(\overline{c}\) is given by the formula:
\[ [\overline{a}, \overline{b}, \overline{c}] = \overline{a} \cdot (\overline{b} \times \overline{c}) \]
In this case, we have \(\overline{c} = \overline{a} \times \overline{b}\).

### Step 2: Substitute the Cross Product
We can substitute \(\overline{c}\) into the scalar triple product:
\[ [\overline{a}, \overline{b}, \overline{a} \times \overline{b}] = \overline{a} \cdot (\overline{b} \times (\overline{a} \times \overline{b})) \]

### Step 3: Use the Vector Triple Product Identity
We can apply the vector triple product identity:
\[ \overline{x} \times (\overline{y} \times \overline{z}) = (\overline{x} \cdot \overline{z}) \overline{y} - (\overline{x} \cdot \overline{y}) \overline{z} \]
Here, let \(\overline{x} = \overline{a}\), \(\overline{y} = \overline{b}\), and \(\overline{z} = \overline{b}\):
\[ \overline{b} \times (\overline{a} \times \overline{b}) = (\overline{b} \cdot \overline{b}) \overline{a} - (\overline{b} \cdot \overline{a}) \overline{b} \]

### Step 4: Simplify the Expression
Now substituting back into our expression:
\[ [\overline{a}, \overline{b}, \overline{a} \times \overline{b}] = \overline{a} \cdot \left( \|\overline{b}\|^2 \overline{a} - (\overline{a} \cdot \overline{b}) \overline{b} \right) \]
This expands to:
\[ \overline{a} \cdot \|\overline{b}\|^2 \overline{a} - \overline{a} \cdot ((\overline{a} \cdot \overline{b}) \overline{b}) \]

### Step 5: Calculate Each Dot Product
The first term simplifies to:
\[ \|\overline{b}\|^2 (\overline{a} \cdot \overline{a}) = \|\overline{b}\|^2 \|\overline{a}\|^2 \]
The second term simplifies to:
\[ (\overline{a} \cdot \overline{b}) (\overline{a} \cdot \overline{b}) = (\overline{a} \cdot \overline{b})^2 \]

### Step 6: Combine the Results
Thus, we have:
\[ [\overline{a}, \overline{b}, \overline{a} \times \overline{b}] = \|\overline{b}\|^2 \|\overline{a}\|^2 - (\overline{a} \cdot \overline{b})^2 \]

### Final Result
The final result of the scalar triple product is:
\[ [\overline{a}, \overline{b}, \overline{a} \times \overline{b}] = \|\overline{b}\|^2 \|\overline{a}\|^2 - (\overline{a} \cdot \overline{b})^2 \]