If `vecA=3hati+4hatj` and `vecB=6hati+8hatj`, select correct alternatives :- `(i) vecA.vecB=50" "(ii)2A=B` `(iii)hatA=hatB" "(iv) hatAxxvecB=vec0`

Answer

To solve the problem, we will analyze the given vectors and check each of the statements provided.

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

### Step 1: Calculate the dot product \(\vec{A} \cdot \vec{B}\)

The dot product of two vectors \(\vec{A} = a_1\hat{i} + a_2\hat{j}\) and \(\vec{B} = b_1\hat{i} + b_2\hat{j}\) is given by:
\[ \vec{A} \cdot \vec{B} = a_1b_1 + a_2b_2 \]

Substituting the values:
\[ \vec{A} \cdot \vec{B} = (3)(6) + (4)(8) = 18 + 32 = 50 \]

### Step 2: Check if \(2\vec{A} = \vec{B}\)

To check if \(2\vec{A} = \vec{B}\), we calculate \(2\vec{A}\):
\[ 2\vec{A} = 2(3\hat{i} + 4\hat{j}) = 6\hat{i} + 8\hat{j} \]
Since \(2\vec{A} = \vec{B}\), this statement is true.

### Step 3: Check if \(\hat{A} = \hat{B}\)

The unit vector \(\hat{A}\) is given by:
\[ \hat{A} = \frac{\vec{A}}{|\vec{A}|} \]
where \(|\vec{A}|\) is the magnitude of \(\vec{A}\):
\[ |\vec{A}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Thus,
\[ \hat{A} = \frac{3\hat{i} + 4\hat{j}}{5} \]

Similarly, for \(\vec{B}\):
\[ |\vec{B}| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]
Thus,
\[ \hat{B} = \frac{6\hat{i} + 8\hat{j}}{10} = \frac{3\hat{i} + 4\hat{j}}{5} \]

Since \(\hat{A} = \hat{B}\), this statement is also true.

### Step 4: Check if \(\hat{A} \times \vec{B} = \vec{0}\)

The cross product \(\hat{A} \times \vec{B}\) can be calculated as follows:
\[ \hat{A} = \frac{3\hat{i} + 4\hat{j}}{5} \]
\[ \vec{B} = 6\hat{i} + 8\hat{j} \]

The cross product of two vectors in the plane (2D) is zero because they are parallel. Therefore:
\[ \hat{A} \times \vec{B} = \vec{0} \]

### Conclusion

The correct alternatives are:
1. \(\vec{A} \cdot \vec{B} = 50\) (True)
2. \(2\vec{A} = \vec{B}\) (True)
3. \(\hat{A} = \hat{B}\) (True)
4. \(\hat{A} \times \vec{B} = \vec{0}\) (True)