If `a_(1) and a_(2)` aare two non- collineaar unit vectors and if `|a_(1)+a_(2)|=sqrt(3),` ,then value of `(a_(1)-a_(2)).(2a_(1)-a_(2))` is

Options

A. 2
B. `(3)/(2)`
C. `(1)/(2)`
D. 1

Answer

To solve the problem, we need to find the value of \((\mathbf{a_1} - \mathbf{a_2}) \cdot (2\mathbf{a_1} - \mathbf{a_2})\) given that \(|\mathbf{a_1} + \mathbf{a_2}| = \sqrt{3}\) and that \(\mathbf{a_1}\) and \(\mathbf{a_2}\) are non-collinear unit vectors.

### Step-by-step Solution:

1. **Understanding the Magnitude Condition:**
We start with the equation given by the magnitude:
\[ |\mathbf{a_1} + \mathbf{a_2}| = \sqrt{3} \]
Squaring both sides gives:
\[ |\mathbf{a_1} + \mathbf{a_2}|^2 = 3 \]

2. **Expanding the Magnitude:**
The magnitude squared can be expanded as follows:
\[ |\mathbf{a_1} + \mathbf{a_2}|^2 = \mathbf{a_1} \cdot \mathbf{a_1} + \mathbf{a_2} \cdot \mathbf{a_2} + 2 \mathbf{a_1} \cdot \mathbf{a_2} \]
Since \(\mathbf{a_1}\) and \(\mathbf{a_2}\) are unit vectors:
\[ \mathbf{a_1} \cdot \mathbf{a_1} = 1 \quad \text{and} \quad \mathbf{a_2} \cdot \mathbf{a_2} = 1 \]
Thus, we have:
\[ 1 + 1 + 2 \mathbf{a_1} \cdot \mathbf{a_2} = 3 \]

3. **Solving for \(\mathbf{a_1} \cdot \mathbf{a_2}\):**
Simplifying the equation gives:
\[ 2 + 2 \mathbf{a_1} \cdot \mathbf{a_2} = 3 \]
Therefore:
\[ 2 \mathbf{a_1} \cdot \mathbf{a_2} = 1 \quad \Rightarrow \quad \mathbf{a_1} \cdot \mathbf{a_2} = \frac{1}{2} \]

4. **Finding the Dot Product:**
Now we need to compute \((\mathbf{a_1} - \mathbf{a_2}) \cdot (2\mathbf{a_1} - \mathbf{a_2})\):
\[ (\mathbf{a_1} - \mathbf{a_2}) \cdot (2\mathbf{a_1} - \mathbf{a_2}) = \mathbf{a_1} \cdot (2\mathbf{a_1} - \mathbf{a_2}) - \mathbf{a_2} \cdot (2\mathbf{a_1} - \mathbf{a_2}) \]

5. **Expanding the Expression:**
Expanding this gives:
\[ = 2 \mathbf{a_1} \cdot \mathbf{a_1} - \mathbf{a_1} \cdot \mathbf{a_2} - 2 \mathbf{a_2} \cdot \mathbf{a_1} + \mathbf{a_2} \cdot \mathbf{a_2} \]
Substituting the known values:
\[ = 2(1) - \frac{1}{2} - 2\left(\frac{1}{2}\right) + 1 \]
Simplifying this:
\[ = 2 - \frac{1}{2} - 1 + 1 = 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \]

6. **Final Result:**
Therefore, the value of \((\mathbf{a_1} - \mathbf{a_2}) \cdot (2\mathbf{a_1} - \mathbf{a_2})\) is:
\[ \frac{3}{2} \]