If `|hata-hatb|=sqrt(2)` then calculate the value of `|hat a+sqrt(3)hatb|`.

Options

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Explanation

To solve the problem, we need to calculate the value of \(|\hat{a} + \sqrt{3} \hat{b}|\) given that \(|\hat{a} - \hat{b}| = \sqrt{2}\).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We are given that \(|\hat{a} - \hat{b}| = \sqrt{2}\). Since \(\hat{a}\) and \(\hat{b}\) are unit vectors, their magnitudes are both equal to 1.

2. **Using the Magnitude Formula**:
The magnitude of the difference of two vectors can be expressed as:
\[ |\hat{a} - \hat{b}| = \sqrt{|\hat{a}|^2 + |\hat{b}|^2 - 2 |\hat{a}| |\hat{b}| \cos \theta} \]
where \(\theta\) is the angle between the vectors \(\hat{a}\) and \(\hat{b}\).

3. **Substituting the Values**:
Since \(|\hat{a}| = 1\) and \(|\hat{b}| = 1\), we can substitute these values into the formula:
\[ |\hat{a} - \hat{b}| = \sqrt{1^2 + 1^2 - 2 \cdot 1 \cdot 1 \cdot \cos \theta} = \sqrt{2 - 2 \cos \theta} \]
Setting this equal to \(\sqrt{2}\):
\[ \sqrt{2 - 2 \cos \theta} = \sqrt{2} \]

4. **Squaring Both Sides**:
Squaring both sides gives:
\[ 2 - 2 \cos \theta = 2 \]
Simplifying this, we find:
\[ -2 \cos \theta = 0 \implies \cos \theta = 0 \]

5. **Finding the Angle**:
Since \(\cos \theta = 0\), the angle \(\theta\) is \(90^\circ\). This means that the vectors \(\hat{a}\) and \(\hat{b}\) are perpendicular to each other.

6. **Calculating \(|\hat{a} + \sqrt{3} \hat{b}|\)**:
Now we need to find the magnitude of \(|\hat{a} + \sqrt{3} \hat{b}|\):
\[ |\hat{a} + \sqrt{3} \hat{b}| = \sqrt{|\hat{a}|^2 + |\sqrt{3} \hat{b}|^2 + 2 |\hat{a}| |\sqrt{3} \hat{b}| \cos \theta} \]
Substituting the known values:
\[ |\hat{a} + \sqrt{3} \hat{b}| = \sqrt{1^2 + (\sqrt{3})^2 + 2 \cdot 1 \cdot \sqrt{3} \cdot 1 \cdot \cos 90^\circ} \]
Since \(\cos 90^\circ = 0\), this simplifies to:
\[ |\hat{a} + \sqrt{3} \hat{b}| = \sqrt{1 + 3 + 0} = \sqrt{4} = 2 \]

### Final Answer:
Thus, the value of \(|\hat{a} + \sqrt{3} \hat{b}|\) is \(2\).