If `hata" and "hatb` are non-collinear unit vectors and if `|hata+hatb|=sqrt(3)`, then the value of `(2hata-5hatb).(3hata+hatb)` is :-

Options

A. `(41)/(2)`
B. `(11)/(2)`
C. `-(11)/(2)`
D. `-(41)/(2)`

Answer

To solve the problem step by step, we start with the given information and apply vector algebra.

### Step 1: Understanding the given information
We have two non-collinear unit vectors \(\hat{a}\) and \(\hat{b}\) such that:
\[ |\hat{a} + \hat{b}| = \sqrt{3} \]

### Step 2: Squaring both sides
To eliminate the square root, we square both sides:
\[ |\hat{a} + \hat{b}|^2 = 3 \]
Using the property of magnitudes, we can expand the left side:
\[ |\hat{a}|^2 + |\hat{b}|^2 + 2 \hat{a} \cdot \hat{b} = 3 \]

### Step 3: Substituting the magnitudes
Since \(\hat{a}\) and \(\hat{b}\) are unit vectors, we know:
\[ |\hat{a}|^2 = 1 \quad \text{and} \quad |\hat{b}|^2 = 1 \]
Substituting these values into the equation gives:
\[ 1 + 1 + 2 \hat{a} \cdot \hat{b} = 3 \]
This simplifies to:
\[ 2 + 2 \hat{a} \cdot \hat{b} = 3 \]

### Step 4: Solving for \(\hat{a} \cdot \hat{b}\)
Rearranging the equation, we find:
\[ 2 \hat{a} \cdot \hat{b} = 3 - 2 \]
\[ 2 \hat{a} \cdot \hat{b} = 1 \]
Thus:
\[ \hat{a} \cdot \hat{b} = \frac{1}{2} \]

### Step 5: Finding the dot product \((2\hat{a} - 5\hat{b}) \cdot (3\hat{a} + \hat{b})\)
We need to compute:
\[ (2\hat{a} - 5\hat{b}) \cdot (3\hat{a} + \hat{b}) \]
Using the distributive property of the dot product:
\[ = 2\hat{a} \cdot 3\hat{a} + 2\hat{a} \cdot \hat{b} - 5\hat{b} \cdot 3\hat{a} - 5\hat{b} \cdot \hat{b} \]

### Step 6: Calculating each term
Calculating each term:
1. \(2\hat{a} \cdot 3\hat{a} = 6\hat{a} \cdot \hat{a} = 6 \cdot 1 = 6\)
2. \(2\hat{a} \cdot \hat{b} = 2 \cdot \frac{1}{2} = 1\)
3. \(-5\hat{b} \cdot 3\hat{a} = -15\hat{b} \cdot \hat{a} = -15 \cdot \frac{1}{2} = -\frac{15}{2}\)
4. \(-5\hat{b} \cdot \hat{b} = -5 \cdot 1 = -5\)

### Step 7: Combining all terms
Now, we combine all the calculated terms:
\[ 6 + 1 - \frac{15}{2} - 5 \]
Combining the constants:
\[ = 7 - \frac{15}{2} - 5 \]
\[ = 7 - 5 - \frac{15}{2} = 2 - \frac{15}{2} = \frac{4}{2} - \frac{15}{2} = -\frac{11}{2} \]

### Final Answer
Thus, the value of \((2\hat{a} - 5\hat{b}) \cdot (3\hat{a} + \hat{b})\) is:
\[ -\frac{11}{2} \]