If `vec(a_(1)) and vec(a_(2))` are two non-collinear unit vectors and if `|vec(a_(1)) + vec(a_(2))|=sqrt(3)`, then the value of `(vec(a_(1))-vec(a_(2))). (2 vec(a_(1))+vec(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 \((\vec{a_1} - \vec{a_2}) \cdot (2\vec{a_1} + \vec{a_2})\) given that \(|\vec{a_1} + \vec{a_2}| = \sqrt{3}\) and that \(\vec{a_1}\) and \(\vec{a_2}\) are non-collinear unit vectors.

### Step-by-Step Solution:

1. **Understanding the Magnitude Condition**:
Since \(\vec{a_1}\) and \(\vec{a_2}\) are unit vectors, we have:
\[ |\vec{a_1}| = 1 \quad \text{and} \quad |\vec{a_2}| = 1 \]
We are given:
\[ |\vec{a_1} + \vec{a_2}| = \sqrt{3} \]

2. **Using the Magnitude Formula**:
The magnitude of the sum of two vectors can be expressed as:
\[ |\vec{a_1} + \vec{a_2}|^2 = |\vec{a_1}|^2 + |\vec{a_2}|^2 + 2\vec{a_1} \cdot \vec{a_2} \]
Substituting the known values:
\[ (\sqrt{3})^2 = 1^2 + 1^2 + 2\vec{a_1} \cdot \vec{a_2} \]
This simplifies to:
\[ 3 = 1 + 1 + 2\vec{a_1} \cdot \vec{a_2} \]
\[ 3 = 2 + 2\vec{a_1} \cdot \vec{a_2} \]
Rearranging gives:
\[ 2\vec{a_1} \cdot \vec{a_2} = 1 \quad \Rightarrow \quad \vec{a_1} \cdot \vec{a_2} = \frac{1}{2} \]

3. **Finding the Required Dot Product**:
We need to calculate:
\[ (\vec{a_1} - \vec{a_2}) \cdot (2\vec{a_1} + \vec{a_2}) \]
Expanding this using the distributive property:
\[ = \vec{a_1} \cdot (2\vec{a_1}) + \vec{a_1} \cdot \vec{a_2} - \vec{a_2} \cdot (2\vec{a_1}) - \vec{a_2} \cdot \vec{a_2} \]
This simplifies to:
\[ = 2(\vec{a_1} \cdot \vec{a_1}) + \vec{a_1} \cdot \vec{a_2} - 2(\vec{a_2} \cdot \vec{a_1}) - (\vec{a_2} \cdot \vec{a_2}) \]
Since \(\vec{a_1} \cdot \vec{a_1} = 1\) and \(\vec{a_2} \cdot \vec{a_2} = 1\):
\[ = 2(1) + \frac{1}{2} - 2\left(\frac{1}{2}\right) - 1 \]
Simplifying this gives:
\[ = 2 + \frac{1}{2} - 1 - 1 \]
\[ = 2 + \frac{1}{2} - 2 = \frac{1}{2} \]

4. **Final Result**:
Therefore, the value of \((\vec{a_1} - \vec{a_2}) \cdot (2\vec{a_1} + \vec{a_2})\) is:
\[ \boxed{\frac{1}{2}} \]