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. exactly two values of `theta`
B. more than two but not all values of `theta`
C. no value of `theta`
D. exactly one value of `theta`

Answer

To determine how many values of \( \theta \) make the vector \( 2\vec{u} \times 3\vec{v} \) a unit vector, we can follow these steps:

### Step 1: Understand the Cross Product
The cross product \( \vec{a} \times \vec{b} \) of two vectors results in a vector that is perpendicular to both \( \vec{a} \) and \( \vec{b} \). The magnitude of the cross product 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 to the Given Vectors
In our case, we have:
\[ \vec{u} \text{ and } \vec{v} \text{ are unit vectors, so } |\vec{u}| = 1 \text{ and } |\vec{v}| = 1. \]
Thus, the magnitude of the cross product \( 2\vec{u} \times 3\vec{v} \) can be calculated as:
\[ |2\vec{u} \times 3\vec{v}| = |2| |\vec{u}| |3| |\vec{v}| \sin \theta = 6 |\vec{u} \times \vec{v}|. \]

### Step 3: 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:
\[ 6 |\vec{u} \times \vec{v}| = 1. \]
This implies:
\[ |\vec{u} \times \vec{v}| = \frac{1}{6}. \]

### Step 4: Substitute the Magnitude of the Cross Product
Since \( |\vec{u} \times \vec{v}| = |\vec{u}| |\vec{v}| \sin \theta \) and both \( \vec{u} \) and \( \vec{v} \) are unit vectors, we have:
\[ |\vec{u} \times \vec{v}| = 1 \cdot 1 \cdot \sin \theta = \sin \theta. \]
Thus, we can write:
\[ \sin \theta = \frac{1}{6}. \]

### Step 5: Determine the Values of \( \theta \)
The sine function is positive in the first and second quadrants. Therefore, the equation \( \sin \theta = \frac{1}{6} \) has two solutions in the range \( [0, 180^\circ] \):
1. \( \theta = \arcsin\left(\frac{1}{6}\right) \) (first quadrant)
2. \( \theta = 180^\circ - \arcsin\left(\frac{1}{6}\right) \) (second quadrant)

### Conclusion
Since \( \theta \) is specified as an acute angle, we only consider the first solution. Thus, \( 2\vec{u} \times 3\vec{v} \) is a unit vector for only one value of \( \theta \).

### Final Answer
The vector \( 2\vec{u} \times 3\vec{v} \) is a unit vector for **one value of \( \theta \)**.

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