`|[overline(a)*overline(a), overline(a)*overline(b)], [overline(a)*overline(b), overline(b)*overline(b)]|=`

Options

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

Answer

To solve the given determinant

\[ D = \begin{vmatrix} \overline{a} \cdot \overline{a} & \overline{a} \cdot \overline{b} \\ \overline{a} \cdot \overline{b} & \overline{b} \cdot \overline{b} \end{vmatrix} \]

we can follow these steps:

### Step 1: Identify the dot products
The elements of the determinant can be expressed in terms of the magnitudes of vectors \(\overline{a}\) and \(\overline{b}\) and the angle \(\theta\) between them.

- \(\overline{a} \cdot \overline{a} = |\overline{a}|^2 = A^2\)
- \(\overline{b} \cdot \overline{b} = |\overline{b}|^2 = B^2\)
- \(\overline{a} \cdot \overline{b} = |\overline{a}| |\overline{b}| \cos \theta = AB \cos \theta\)

### Step 2: Substitute into the determinant
Substituting these values into the determinant gives us:

\[ D = \begin{vmatrix} A^2 & AB \cos \theta \\ AB \cos \theta & B^2 \end{vmatrix} \]

### Step 3: Calculate the determinant
Using the formula for the determinant of a 2x2 matrix, we have:

\[ D = A^2 \cdot B^2 - (AB \cos \theta)^2 \]

### Step 4: Simplify the expression
Now, simplifying the expression:

\[ D = A^2 B^2 - A^2 B^2 \cos^2 \theta \]

Factoring out \(A^2 B^2\):

\[ D = A^2 B^2 (1 - \cos^2 \theta) \]

### Step 5: Use the trigonometric identity
Using the identity \(1 - \cos^2 \theta = \sin^2 \theta\):

\[ D = A^2 B^2 \sin^2 \theta \]

### Step 6: Express in terms of the cross product
The magnitude of the cross product of two vectors \(\overline{a}\) and \(\overline{b}\) is given by:

\[ |\overline{a} \times \overline{b}| = AB \sin \theta \]

Thus, we can rewrite \(D\) as:

\[ D = (AB \sin \theta)^2 \]

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

\[ D = |\overline{a} \times \overline{b}|^2 \]