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

Options

A. (i, ii)
B. (ii, iii)
C. (i, iv)
D. (i, ii, iii, iv)

Answer

To solve the problem, we need to analyze the given vectors \(\vec{A}\) and \(\vec{B}\) and evaluate the provided statements one by one.

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

**Conclusion for (i)**: \(\vec{A} \cdot \vec{B} = 50\) is **correct**.

### Step 2: Check if \(2\vec{A} = \vec{B}\)
To check this, we can multiply \(\vec{A}\) by 2:
\[ 2\vec{A} = 2(3\hat{i} + 4\hat{j}) = 6\hat{i} + 8\hat{j} = \vec{B} \]

**Conclusion for (ii)**: \(2\vec{A} = \vec{B}\) is **correct**.

### Step 3: Find the unit vectors \(\hat{A}\) and \(\hat{B}\)
The unit vector \(\hat{A}\) is given by:
\[ \hat{A} = \frac{\vec{A}}{|\vec{A}|} \]
First, we calculate the magnitude of \(\vec{A}\):
\[ |\vec{A}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Thus,
\[ \hat{A} = \frac{3\hat{i} + 4\hat{j}}{5} = \frac{3}{5}\hat{i} + \frac{4}{5}\hat{j} \]

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

Since \(\hat{A} = \hat{B}\), we conclude that:

**Conclusion for (iii)**: \(\hat{A} = \hat{B}\) is **correct**.

### Step 4: Check if \(\hat{A} \times \hat{B} = \vec{0}\)
The cross product of two vectors is zero if they are parallel (or in the same direction). Since we found that \(\hat{A} = \hat{B}\), it follows that:
\[ \hat{A} \times \hat{B} = \vec{0} \]

**Conclusion for (iv)**: \(\hat{A} \times \hat{B} = \vec{0}\) is **correct**.

### Final Conclusion:
All statements (i), (ii), (iii), and (iv) are correct.

### Summary of Correct Alternatives:
- (i) \(\vec{A} \cdot \vec{B} = 50\)
- (ii) \(2\vec{A} = \vec{B}\)
- (iii) \(\hat{A} = \hat{B}\)
- (iv) \(\hat{A} \times \hat{B} = \vec{0}\)