Given `A= 3hati+4hatj and B=6hati+8hatj ` which of the following statement is correct ?

Options

A. `AxxB=0`
B. `(|A|)/(|B|)=(1)/(2)`
C. `|A|=15`
D. `A.|B|=48`

Explanation

To solve the problem, we need to analyze the vectors \( A \) and \( B \) given as:

\[ A = 3\hat{i} + 4\hat{j} \]
\[ B = 6\hat{i} + 8\hat{j} \]

### Step 1: Calculate the Cross Product \( A \times B \)

The cross product of two vectors \( A \) and \( B \) can be calculated using the determinant of a matrix formed by the unit vectors \( \hat{i}, \hat{j}, \hat{k} \) and the components of the vectors \( A \) and \( B \).

\[ A \times B = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 4 & 0 \\ 6 & 8 & 0 \end{vmatrix} \]

### Step 2: Calculate the Determinant

To calculate the determinant, we can expand it as follows:

\[ A \times B = \hat{i} \begin{vmatrix} 4 & 0 \\ 8 & 0 \end{vmatrix} - \hat{j} \begin{vmatrix} 3 & 0 \\ 6 & 0 \end{vmatrix} + \hat{k} \begin{vmatrix} 3 & 4 \\ 6 & 8 \end{vmatrix} \]

Calculating each of these determinants:

1. For \( \hat{i} \):
\[ \begin{vmatrix} 4 & 0 \\ 8 & 0 \end{vmatrix} = (4 \cdot 0) - (0 \cdot 8) = 0 \]

2. For \( \hat{j} \):
\[ \begin{vmatrix} 3 & 0 \\ 6 & 0 \end{vmatrix} = (3 \cdot 0) - (0 \cdot 6) = 0 \]

3. For \( \hat{k} \):
\[ \begin{vmatrix} 3 & 4 \\ 6 & 8 \end{vmatrix} = (3 \cdot 8) - (4 \cdot 6) = 24 - 24 = 0 \]

### Step 3: Combine the Results

Putting it all together:

\[ A \times B = 0\hat{i} - 0\hat{j} + 0\hat{k} = 0 \]

### Conclusion for Cross Product

Since \( A \times B = 0 \), this means that vectors \( A \) and \( B \) are parallel.

### Step 4: Calculate the Magnitudes of \( A \) and \( B \)

1. Magnitude of \( A \):
\[ |A| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

2. Magnitude of \( B \):
\[ |B| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]

### Step 5: Ratio of Magnitudes

Now, we can find the ratio of the magnitudes of \( A \) and \( B \):
\[ \frac{|A|}{|B|} = \frac{5}{10} = \frac{1}{2} \]

### Summary of Findings

1. \( A \times B = 0 \) (indicating \( A \) and \( B \) are parallel).
2. \( \frac{|A|}{|B|} = \frac{1}{2} \).

### Final Answer

The correct statements based on the calculations are:
- \( A \times B = 0 \)
- \( \frac{|A|}{|B|} = \frac{1}{2} \)