If `vecA=3hati+4hatj` and `vecB=6hati+8hatj` and A and B are the magnitudes of `vecA` and `vecB`, then which of the following is not true?

Options

A. `vecAxx vecB= vec0`
B. `(A)/(B) = (1)/(2)`
C. `vecA*vecB= 48`
D. `A= 5`

Explanation

To solve the problem, we need to analyze the vectors \(\vec{A}\) and \(\vec{B}\), calculate their magnitudes, and then evaluate the given statements to find out which one is not true.

### Step-by-Step Solution:

1. **Define the Vectors:**
\[ \vec{A} = 3\hat{i} + 4\hat{j} \]
\[ \vec{B} = 6\hat{i} + 8\hat{j} \]

2. **Calculate the Magnitude of Vector A:**
The magnitude of vector \(\vec{A}\) is given by:
\[ A = |\vec{A}| = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

3. **Calculate the Magnitude of Vector B:**
The magnitude of vector \(\vec{B}\) is given by:
\[ B = |\vec{B}| = \sqrt{(6)^2 + (8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]

4. **Evaluate the Cross Product \(\vec{A} \times \vec{B}\):**
The cross product can be calculated using the determinant of a matrix:
\[ \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 4 & 0 \\ 6 & 8 & 0 \end{vmatrix} \]
\[ = \hat{i}(4 \cdot 0 - 0 \cdot 8) - \hat{j}(3 \cdot 0 - 0 \cdot 6) + \hat{k}(3 \cdot 8 - 4 \cdot 6) \]
\[ = 0\hat{i} - 0\hat{j} + (24 - 24)\hat{k} = 0\hat{k} \]
Thus, \(\vec{A} \times \vec{B} = \vec{0}\) (zero vector).

5. **Evaluate the Dot Product \(\vec{A} \cdot \vec{B}\):**
The dot product is calculated as follows:
\[ \vec{A} \cdot \vec{B} = (3\hat{i} + 4\hat{j}) \cdot (6\hat{i} + 8\hat{j}) \]
\[ = 3 \cdot 6 + 4 \cdot 8 = 18 + 32 = 50 \]

6. **Analyze the Options:**
- Option A: \(\vec{A} \times \vec{B} = \vec{0}\) (True)
- Option B: \(\frac{A}{B} = \frac{5}{10} = \frac{1}{2}\) (True)
- Option C: \(\vec{A} \cdot \vec{B} = 48\) (False, we found it to be 50)
- Option D: \(A = 5\) (True)

### Conclusion:
The statement that is **not true** is Option C, which claims that \(\vec{A} \cdot \vec{B} = 48\).