Given `vec(A)=4hat(i)+6hat(j)` and `vec(B)=2hat(i)+3hat(j)`. Which of the following

Options

A. `vec(A)xxvec(B)=vec(0)`
B. `vec(A).vec(B)=24`
C. `(|vec(A)|)/(|vec(B)|)=1/2`
D. `vec(A)` and `vec(B)` are antiparallel

Explanation

To solve the problem step by step, we will analyze the vectors \(\vec{A}\) and \(\vec{B}\) given in the question.

### Step 1: Write down the vectors
Given:
\[ \vec{A} = 4\hat{i} + 6\hat{j} \]
\[ \vec{B} = 2\hat{i} + 3\hat{j} \]

### Step 2: Identify the relationship between the vectors
We can express \(\vec{A}\) in terms of \(\vec{B}\):
\[ \vec{A} = 2 \cdot (2\hat{i} + 3\hat{j}) = 2\vec{B} \]
This shows that \(\vec{A}\) is a scalar multiple of \(\vec{B}\).

### Step 3: Determine if the vectors are parallel
Since \(\vec{A} = 2\vec{B}\), it indicates that both vectors are parallel. Parallel vectors have the same direction.

### Step 4: Calculate the cross product
The cross product of two parallel vectors is zero:
\[ \vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin(\theta) \]
Since \(\theta = 0^\circ\) (the angle between parallel vectors), we have:
\[ \sin(0^\circ) = 0 \]
Thus,
\[ \vec{A} \times \vec{B} = 0 \]

### Step 5: Calculate the dot product
The dot product of \(\vec{A}\) and \(\vec{B}\) can be calculated as follows:
\[ \vec{A} \cdot \vec{B} = (4\hat{i} + 6\hat{j}) \cdot (2\hat{i} + 3\hat{j}) = (4 \cdot 2) + (6 \cdot 3) = 8 + 18 = 26 \]

### Step 6: Magnitude of the vectors
Calculate the magnitudes of \(\vec{A}\) and \(\vec{B}\):
\[ |\vec{A}| = \sqrt{(4^2 + 6^2)} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13} \]
\[ |\vec{B}| = \sqrt{(2^2 + 3^2)} = \sqrt{4 + 9} = \sqrt{13} \]
We can see that:
\[ |\vec{A}| = 2 |\vec{B}| \]

### Conclusion
From the calculations, we conclude that:
- \(\vec{A}\) and \(\vec{B}\) are parallel vectors.
- The cross product \(\vec{A} \times \vec{B} = 0\).
- The dot product \(\vec{A} \cdot \vec{B} = 26\).

### Final Answer
The correct option is that \(\vec{A}\) and \(\vec{B}\) are parallel.