Consider the following statements in respect of the vectors `vec(u_(1))=(1, 2, 3), vec(u_(2))=(2, 3, 1), vec(u_(3))=(1, 3, 2) and vec(u_(4))=(4, 6, 2)`
I. `vec(u_(1)) ` is parallel to `vec(u_(4))`.
II. `vec(u_(2))` is parallel to `vec(u_(4))`.
III. `vec(u_(2))` is parallel to `vec(u_(3))`.
Which of the statements given above is/are correct?

Options

A. Only I
B. Only II
C. Only III
D. Both I and III

Explanation

To determine which of the statements regarding the vectors \( \vec{u_1} = (1, 2, 3) \), \( \vec{u_2} = (2, 3, 1) \), \( \vec{u_3} = (1, 3, 2) \), and \( \vec{u_4} = (4, 6, 2) \) are correct, we need to check the conditions for parallelism between the vectors.

### Step-by-Step Solution

1. **Check if \( \vec{u_1} \) is parallel to \( \vec{u_4} \)**:
- Two vectors \( \vec{a} \) and \( \vec{b} \) are parallel if there exists a scalar \( k \) such that \( \vec{a} = k \vec{b} \).
- Here, we compare \( \vec{u_1} = (1, 2, 3) \) and \( \vec{u_4} = (4, 6, 2) \).
- Let's find \( k \) for each component:
- For the first component: \( 1k = 4 \) → \( k = 4 \)
- For the second component: \( 2k = 6 \) → \( k = 3 \)
- For the third component: \( 3k = 2 \) → \( k = \frac{2}{3} \)
- Since \( k \) is not the same for all components, \( \vec{u_1} \) is **not parallel** to \( \vec{u_4} \).

2. **Check if \( \vec{u_2} \) is parallel to \( \vec{u_4} \)**:
- Compare \( \vec{u_2} = (2, 3, 1) \) and \( \vec{u_4} = (4, 6, 2) \).
- Find \( k \):
- For the first component: \( 2k = 4 \) → \( k = 2 \)
- For the second component: \( 3k = 6 \) → \( k = 2 \)
- For the third component: \( 1k = 2 \) → \( k = 2 \)
- Since \( k \) is the same for all components, \( \vec{u_2} \) is **parallel** to \( \vec{u_4} \).

3. **Check if \( \vec{u_2} \) is parallel to \( \vec{u_3} \)**:
- Compare \( \vec{u_2} = (2, 3, 1) \) and \( \vec{u_3} = (1, 3, 2) \).
- Find \( k \):
- For the first component: \( 2k = 1 \) → \( k = \frac{1}{2} \)
- For the second component: \( 3k = 3 \) → \( k = 1 \)
- For the third component: \( 1k = 2 \) → \( k = 2 \)
- Since \( k \) is not the same for all components, \( \vec{u_2} \) is **not parallel** to \( \vec{u_3} \).

### Conclusion
- Statement I: \( \vec{u_1} \) is parallel to \( \vec{u_4} \) - **False**
- Statement II: \( \vec{u_2} \) is parallel to \( \vec{u_4} \) - **True**
- Statement III: \( \vec{u_2} \) is parallel to \( \vec{u_3} \) - **False**

Thus, the only correct statement is **II**.