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

Options

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

Answer

To solve the problem step by step, 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 1: Understand the Cross Product
The cross product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by:
\[ \|\vec{a} \times \vec{b}\| = \|\vec{a}\| \|\vec{b}\| \sin \theta \]
where \( \theta \) is the angle between the two vectors.

### Step 2: Apply the Cross Product Formula
In our case, we have:
\[ \vec{a} = 2\vec{u} \quad \text{and} \quad \vec{b} = 3\vec{v} \]
Thus, the magnitude of the cross product is:
\[ \|2\vec{u} \times 3\vec{v}\| = \|2\vec{u}\| \|3\vec{v}\| \sin \theta \]

### Step 3: Calculate the Magnitudes
Since \( \vec{u} \) and \( \vec{v} \) are unit vectors:
\[ \|\vec{u}\| = 1 \quad \text{and} \quad \|\vec{v}\| = 1 \]
Therefore:
\[ \|2\vec{u}\| = 2 \quad \text{and} \quad \|3\vec{v}\| = 3 \]
Substituting these values into the cross product formula gives:
\[ \|2\vec{u} \times 3\vec{v}\| = 2 \cdot 3 \cdot \sin \theta = 6 \sin \theta \]

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

### Step 5: Solve for \( \sin \theta \)
From the equation above, we can solve for \( \sin \theta \):
\[ \sin \theta = \frac{1}{6} \]

### Step 6: Determine the Angle \( \theta \)
Since \( \theta \) is acute, we can find \( \theta \) using the inverse sine function:
\[ \theta = \sin^{-1}\left(\frac{1}{6}\right) \]

### Conclusion
Thus, there is exactly one value of \( \theta \) that satisfies the condition that \( 2\vec{u} \times 3\vec{v} \) is a unit vector.

### Final Answer
Hence, the answer is that there is exactly one value for \( \theta \).

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