If `| vec(a) | = | vec( b) | =1` and `| vec(a ) xx vec( b)| =1`, then

Options

A. `vec(a) _|_vec(b)`
B. `vec(a) ||vec(b)`
C. `vec(a) . vec(b) =1`
D. none of these

Answer

To solve the problem, we need to analyze the given conditions step by step.

### Given:
1. \( |\vec{a}| = |\vec{b}| = 1 \) (both vectors are unit vectors)
2. \( |\vec{a} \times \vec{b}| = 1 \)

### Step 1: Understanding the Cross Product
The magnitude of the cross product of two vectors is given by the formula:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \]
where \( \theta \) is the angle between the two vectors \( \vec{a} \) and \( \vec{b} \).

### Step 2: Substitute the Given Values
Since both \( \vec{a} \) and \( \vec{b} \) are unit vectors, we can substitute their magnitudes into the equation:
\[ |\vec{a} \times \vec{b}| = 1 \cdot 1 \cdot \sin \theta \]
This simplifies to:
\[ |\vec{a} \times \vec{b}| = \sin \theta \]

### Step 3: Set Up the Equation
From the given information, we know:
\[ \sin \theta = 1 \]

### Step 4: Solve for \( \theta \)
The equation \( \sin \theta = 1 \) implies that:
\[ \theta = \frac{\pi}{2} \text{ (or } 90^\circ\text{)} \]
This means that the angle between the vectors \( \vec{a} \) and \( \vec{b} \) is \( 90^\circ \).

### Step 5: Conclusion
Since the angle between \( \vec{a} \) and \( \vec{b} \) is \( 90^\circ \), we can conclude that:
\[ \vec{a} \text{ is perpendicular to } \vec{b} \]

### Final Answer
The correct option is: **A is perpendicular to B.**

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