If `|hatAxxhatB|=-sqrt(3)hatA.hatB`, then `|hatA-hatB|`=…………………..

Options

A. 0
B. 1
C. -1
D. `sqrt(3)`

Answer

To solve the problem \( |\hat{A} - \hat{B}| \) given that \( |\hat{A} \times \hat{B}| = -\sqrt{3} \hat{A} \cdot \hat{B} \), we can follow these steps:

### Step 1: Understand the given equation
The equation states that the magnitude of the cross product of two unit vectors \( \hat{A} \) and \( \hat{B} \) is equal to the negative of the square root of three times their dot product.

### Step 2: Use properties of vectors
Recall the formulas for the cross product and dot product:
- The magnitude of the cross product:
\[ |\hat{A} \times \hat{B}| = |\hat{A}||\hat{B}|\sin\theta \]
- The dot product:
\[ \hat{A} \cdot \hat{B} = |\hat{A}||\hat{B}|\cos\theta \]
Since \( \hat{A} \) and \( \hat{B} \) are unit vectors, their magnitudes are 1. Thus, we can simplify these formulas:
- \( |\hat{A} \times \hat{B}| = \sin\theta \)
- \( \hat{A} \cdot \hat{B} = \cos\theta \)

### Step 3: Set up the equation
From the given condition:
\[ \sin\theta = -\sqrt{3} \cos\theta \]
Dividing both sides by \( \cos\theta \) (assuming \( \cos\theta \neq 0 \)):
\[ \tan\theta = -\sqrt{3} \]

### Step 4: Determine the angle \( \theta \)
The tangent function is negative in the second quadrant. Therefore, the angle \( \theta \) that satisfies \( \tan\theta = -\sqrt{3} \) is:
\[ \theta = 120^\circ \quad \text{or} \quad \theta = \frac{2\pi}{3} \text{ radians} \]

### Step 5: Calculate \( |\hat{A} - \hat{B}| \)
Using the formula for the magnitude of the difference of two vectors:
\[ |\hat{A} - \hat{B}| = \sqrt{|\hat{A}|^2 + |\hat{B}|^2 - 2|\hat{A}||\hat{B}|\cos\theta} \]
Substituting the values:
\[ |\hat{A} - \hat{B}| = \sqrt{1^2 + 1^2 - 2 \cdot 1 \cdot 1 \cdot \cos(120^\circ)} \]
Since \( \cos(120^\circ) = -\frac{1}{2} \):
\[ |\hat{A} - \hat{B}| = \sqrt{1 + 1 - 2 \cdot (-\frac{1}{2})} \]
\[ = \sqrt{2 + 1} = \sqrt{3} \]

### Final Answer
Thus, the magnitude \( |\hat{A} - \hat{B}| \) is:
\[ |\hat{A} - \hat{B}| = \sqrt{3} \]