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)_(3)` is parallel to `vec(u)_(4)`
Which of the statements given above are correct?

Options

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

Answer

To determine which of the statements regarding the vectors \( \vec{u}_1, \vec{u}_2, \vec{u}_3, \) and \( \vec{u}_4 \) are correct, we need to check if any of the vectors are parallel to \( \vec{u}_4 \).

### Given Vectors:
- \( \vec{u}_1 = (1, 2, 3) \)
- \( \vec{u}_2 = (2, 3, 1) \)
- \( \vec{u}_3 = (1, 3, 2) \)
- \( \vec{u}_4 = (4, 6, 2) \)

### Step 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 \( \lambda \) such that:

\[ \vec{a} = \lambda \vec{b} \]

For \( \vec{u}_1 \) and \( \vec{u}_4 \):

Let’s express \( \vec{u}_4 \) in terms of \( \vec{u}_1 \):

\[ \vec{u}_4 = (4, 6, 2) \]

We can check if there is a scalar \( \lambda \) such that:

\[ (1, 2, 3) = \lambda (4, 6, 2) \]

This gives us the equations:

1. \( 1 = 4\lambda \)
2. \( 2 = 6\lambda \)
3. \( 3 = 2\lambda \)

From the first equation:

\[ \lambda = \frac{1}{4} \]

From the second equation:

\[ \lambda = \frac{2}{6} = \frac{1}{3} \]

From the third equation:

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

Since we have different values for \( \lambda \), \( \vec{u}_1 \) is **not parallel** to \( \vec{u}_4\).

### Step 2: Check if \( \vec{u}_2 \) is parallel to \( \vec{u}_4 \)

We check if:

\[ (2, 3, 1) = \lambda (4, 6, 2) \]

This gives us the equations:

1. \( 2 = 4\lambda \)
2. \( 3 = 6\lambda \)
3. \( 1 = 2\lambda \)

From the first equation:

\[ \lambda = \frac{2}{4} = \frac{1}{2} \]

From the second equation:

\[ \lambda = \frac{3}{6} = \frac{1}{2} \]

From the third equation:

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

Since all equations yield the same \( \lambda \), \( \vec{u}_2 \) is **parallel** to \( \vec{u}_4\).

### Step 3: Check if \( \vec{u}_3 \) is parallel to \( \vec{u}_4 \)

We check if:

\[ (1, 3, 2) = \lambda (4, 6, 2) \]

This gives us the equations:

1. \( 1 = 4\lambda \)
2. \( 3 = 6\lambda \)
3. \( 2 = 2\lambda \)

From the first equation:

\[ \lambda = \frac{1}{4} \]

From the second equation:

\[ \lambda = \frac{3}{6} = \frac{1}{2} \]

From the third equation:

\[ \lambda = 1 \]

Since we have different values for \( \lambda \), \( \vec{u}_3 \) is **not parallel** to \( \vec{u}_4\).

### Conclusion:

- Statement I: \( \vec{u}_1 \) is **not parallel** to \( \vec{u}_4\).
- Statement II: \( \vec{u}_2 \) is **parallel** to \( \vec{u}_4\).
- Statement III: \( \vec{u}_3 \) is **not parallel** to \( \vec{u}_4\).

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

### Final Answer:
**Only Statement II is correct.**

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