Statement I: `a=hati+phatj+2hatk and b=2hati+3hatj+qhatk` are parallel vectors, iff `p=(3)/(2) and q=4`.
Statement II: `a=a_(1)hati+a_(2)hatj+a_(3)hatk and b=b_(1)hati+b_(2)hatj+b_(3)hatk` are parallel `(a_(1))/(b_(1))=(a_(2))/(b_(2))=(a_(3))/(b_(3))`.

Options

A. Both Statement I and Statement II are correct and statement II is the correct explanation of statement I
B. Both statement I and statement II are correct but statement II is not the correct explanation of statement I
C. Statement I is correct but statement II is incorrect
D. Statement II is correct but statement I is incorrect

Explanation

To determine whether the statements about the vectors \( \mathbf{a} \) and \( \mathbf{b} \) are true, we will analyze both statements step by step.

### Given:
- \( \mathbf{a} = \hat{i} + p\hat{j} + 2\hat{k} \)
- \( \mathbf{b} = 2\hat{i} + 3\hat{j} + q\hat{k} \)

### Statement I:
The vectors \( \mathbf{a} \) and \( \mathbf{b} \) are parallel if the ratios of their corresponding components are equal. This means:

\[ \frac{1}{2} = \frac{p}{3} = \frac{2}{q} \]

#### Step 1: Set up the first equation
From the first part of the ratio:

\[ \frac{1}{2} = \frac{p}{3} \]

Cross-multiplying gives:

\[ 1 \cdot 3 = 2p \implies 3 = 2p \implies p = \frac{3}{2} \]

**Hint:** To find \( p \), cross-multiply the fractions and solve for \( p \).

#### Step 2: Set up the second equation
Now, using the second part of the ratio:

\[ \frac{p}{3} = \frac{2}{q} \]

Substituting \( p = \frac{3}{2} \) into the equation:

\[ \frac{\frac{3}{2}}{3} = \frac{2}{q} \]

This simplifies to:

\[ \frac{1}{2} = \frac{2}{q} \]

Cross-multiplying gives:

\[ 1 \cdot q = 2 \cdot 2 \implies q = 4 \]

**Hint:** Substitute the value of \( p \) into the second ratio and solve for \( q \) by cross-multiplying.

### Conclusion for Statement I:
Thus, we have found that for the vectors to be parallel, \( p = \frac{3}{2} \) and \( q = 4 \). Therefore, Statement I is correct.

### Statement II:
Statement II states that two vectors \( \mathbf{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \mathbf{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \) are parallel if:

\[ \frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3} \]

This is a known condition for parallel vectors.

### Conclusion for Statement II:
Since this is a standard condition for parallel vectors, Statement II is also correct.

### Final Conclusion:
Both Statement I and Statement II are correct, and Statement II is a correct explanation of Statement I.

### Final Answer:
Both statements are correct, and Statement II is the correct explanation of Statement I.