Two unit vectors under vector multiplication have magnitude one if angle between them is of

Options

A. `45^(@)`
B. `90^(@)`
C. `60^(@)`
D. `120^(@)`

Explanation

To solve the problem, we need to determine the angle between two unit vectors when their vector multiplication (cross product) has a magnitude of 1. Here’s the step-by-step solution:

### Step 1: Understand the Definition of Unit Vectors
We are given two unit vectors, which means that the magnitudes of both vectors \( \mathbf{A} \) and \( \mathbf{B} \) are equal to 1:
\[ |\mathbf{A}| = 1 \quad \text{and} \quad |\mathbf{B}| = 1 \]

**Hint:** Remember that a unit vector has a magnitude of 1.

### Step 2: Recall the Formula for Vector Cross Product
The magnitude of the cross product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by the formula:
\[ |\mathbf{A} \times \mathbf{B}| = |\mathbf{A}| \cdot |\mathbf{B}| \cdot \sin(\theta) \]
where \( \theta \) is the angle between the two vectors.

**Hint:** The cross product involves both the magnitudes of the vectors and the sine of the angle between them.

### Step 3: Substitute the Magnitudes into the Formula
Since both vectors are unit vectors, we can substitute their magnitudes into the equation:
\[ |\mathbf{A} \times \mathbf{B}| = 1 \cdot 1 \cdot \sin(\theta) = \sin(\theta) \]

**Hint:** Simplifying the equation helps to focus on the sine function.

### Step 4: Set Up the Equation
According to the problem, the magnitude of the cross product is given as 1:
\[ \sin(\theta) = 1 \]

**Hint:** This equation tells us about the relationship between the sine of the angle and its maximum value.

### Step 5: Solve for the Angle
The sine function equals 1 at a specific angle:
\[ \sin(\theta) = 1 \implies \theta = 90^\circ \]

**Hint:** Recall the values of sine for common angles to find the solution.

### Conclusion
Thus, the angle between the two unit vectors when their vector multiplication has a magnitude of 1 is:
\[ \theta = 90^\circ \]

**Final Answer:** The angle between the two unit vectors is \( 90^\circ \).

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