If ` vecu and vecv ` are unit vectors and `theta` is the acute angle between them, then ` 2 vecu xx 3vecv` is a unit vector for

Options

A. no value of ` theta`
B. exactly on value of `theta`
C. exactly two values of ` theta`
D. more than two values of ` theta`

Explanation

To solve the problem, we need to determine the conditions under which the vector \( 2\vec{u} \times 3\vec{v} \) is a unit vector, given that \( \vec{u} \) and \( \vec{v} \) are unit vectors and \( \theta \) is the acute angle between them.

### Step-by-Step Solution:

1. **Understanding Unit Vectors**:
Since \( \vec{u} \) and \( \vec{v} \) are unit vectors, we have:
\[ |\vec{u}| = 1 \quad \text{and} \quad |\vec{v}| = 1 \]

2. **Cross Product of Two Vectors**:
The magnitude of the cross product of two vectors is given by:
\[ |\vec{u} \times \vec{v}| = |\vec{u}| |\vec{v}| \sin \theta \]
Substituting the magnitudes of \( \vec{u} \) and \( \vec{v} \):
\[ |\vec{u} \times \vec{v}| = 1 \cdot 1 \cdot \sin \theta = \sin \theta \]

3. **Calculating the Cross Product**:
Now, we need to find the magnitude of \( 2\vec{u} \times 3\vec{v} \):
\[ 2\vec{u} \times 3\vec{v} = 6(\vec{u} \times \vec{v}) \]
Therefore, the magnitude is:
\[ |2\vec{u} \times 3\vec{v}| = 6 |\vec{u} \times \vec{v}| = 6 \sin \theta \]

4. **Condition for a Unit Vector**:
For \( 2\vec{u} \times 3\vec{v} \) to be a unit vector, its magnitude must equal 1:
\[ 6 \sin \theta = 1 \]

5. **Solving for \( \sin \theta \)**:
Rearranging the equation gives:
\[ \sin \theta = \frac{1}{6} \]

6. **Finding the Angle \( \theta \)**:
Since \( \theta \) is an acute angle, we can find the angle using the inverse sine function:
\[ \theta = \arcsin\left(\frac{1}{6}\right) \]

7. **Conclusion**:
Since \( \sin \theta = \frac{1}{6} \) has a unique solution in the interval \( (0, \frac{\pi}{2}) \), we conclude that there is exactly one value of \( \theta \) that satisfies the condition.

### Final Answer:
The answer is that there is exactly one value of \( \theta \) for which \( 2\vec{u} \times 3\vec{v} \) is a unit vector.

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