Let ` vecu = u_(1)hati + u_(2)hatj +u_(3)hatk` be a unit vector in ` R^(3) and vecw = 1/sqrt6 ( hati + hatj + 2hatk)` , Given that there exists a vector `vecv " in " R^(3)` such that ` | vecu xx vecv| =1 and vecw . ( vecu xx vecv) =1` which of the following statements is correct ?

Options

A. There is exactly one choice for such ` vecv`
B. There are exactly two for such `vecv`
C. There are exactly four such `vecv`
D. There are infinitely many choices for such `vecv`

Explanation

To solve the problem step by step, we will analyze the given conditions and derive the necessary conclusions.

### Step 1: Understand the given vectors
We are given:
- \( \vec{u} = u_1 \hat{i} + u_2 \hat{j} + u_3 \hat{k} \) is a unit vector in \( \mathbb{R}^3 \).
- \( \vec{w} = \frac{1}{\sqrt{6}} (\hat{i} + \hat{j} + 2\hat{k}) \).

Since \( \vec{u} \) is a unit vector, we have:
\[ |\vec{u}| = \sqrt{u_1^2 + u_2^2 + u_3^2} = 1. \]

### Step 2: Analyze the conditions
We are given two conditions:
1. \( |\vec{u} \times \vec{v}| = 1 \)
2. \( \vec{w} \cdot (\vec{u} \times \vec{v}) = 1 \)

### Step 3: Use the first condition
From the first condition \( |\vec{u} \times \vec{v}| = 1 \), we know that the magnitude of the cross product of \( \vec{u} \) and \( \vec{v} \) is equal to 1. The magnitude of the cross product can be expressed as:
\[ |\vec{u} \times \vec{v}| = |\vec{u}| |\vec{v}| \sin \theta, \]
where \( \theta \) is the angle between \( \vec{u} \) and \( \vec{v} \). Since \( |\vec{u}| = 1 \), we have:
\[ |\vec{v}| \sin \theta = 1. \]

### Step 4: Use the second condition
From the second condition \( \vec{w} \cdot (\vec{u} \times \vec{v}) = 1 \), we can substitute \( \vec{w} \) and analyze the dot product:
\[ \frac{1}{\sqrt{6}} (\hat{i} + \hat{j} + 2\hat{k}) \cdot (\vec{u} \times \vec{v}) = 1. \]
This implies:
\[ \hat{i} + \hat{j} + 2\hat{k} \cdot (\vec{u} \times \vec{v}) = \sqrt{6}. \]

### Step 5: Relate the two conditions
From the first condition, we have \( |\vec{v}| \sin \theta = 1 \). This means that \( |\vec{v}| \) must be at least \( \frac{1}{\sin \theta} \).

### Step 6: Conclusion
Since both conditions must hold, we conclude that there are infinitely many vectors \( \vec{v} \) that satisfy these conditions, as the angle \( \theta \) can vary while still satisfying \( |\vec{u} \times \vec{v}| = 1 \) and \( \vec{w} \cdot (\vec{u} \times \vec{v}) = 1 \).

### Final Answer
The correct statement is that there are infinitely many vectors \( \vec{v} \) that satisfy the given conditions.