`vecP and vecQ` are two vectors having magnitude 1 and 3 respectively and inclined at an angle `theta` .
If `|vecP+vecQ|=sqrt7`, then find the value of `(2vecP+vecQ).(vecP-vecQ)`

Answer

To solve the problem, we need to find the value of \((2\vec{P} + \vec{Q}) \cdot (\vec{P} - \vec{Q})\). Let's break it down step by step.

### Step 1: Understand the Dot Product
The dot product of two vectors \(\vec{A}\) and \(\vec{B}\) is given by:
\[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\phi) \]
where \(\phi\) is the angle between the two vectors.

### Step 2: Expand the Expression
We can expand \((2\vec{P} + \vec{Q}) \cdot (\vec{P} - \vec{Q})\) using the distributive property of the dot product:
\[ (2\vec{P} + \vec{Q}) \cdot (\vec{P} - \vec{Q}) = 2\vec{P} \cdot \vec{P} - 2\vec{P} \cdot \vec{Q} + \vec{Q} \cdot \vec{P} - \vec{Q} \cdot \vec{Q} \]

### Step 3: Calculate Each Dot Product
1. **Calculate \(\vec{P} \cdot \vec{P}\)**:
\[ \vec{P} \cdot \vec{P} = |\vec{P}|^2 = 1^2 = 1 \]

2. **Calculate \(\vec{Q} \cdot \vec{Q}\)**:
\[ \vec{Q} \cdot \vec{Q} = |\vec{Q}|^2 = 3^2 = 9 \]

3. **Calculate \(\vec{P} \cdot \vec{Q}\)**:
We know from the problem that:
\[ |\vec{P} + \vec{Q}| = \sqrt{7} \]
Therefore,
\[ |\vec{P} + \vec{Q}|^2 = 7 \]
Using the formula for the magnitude of the sum of two vectors:
\[ |\vec{P} + \vec{Q}|^2 = |\vec{P}|^2 + |\vec{Q}|^2 + 2(\vec{P} \cdot \vec{Q}) \]
Substituting the values:
\[ 7 = 1 + 9 + 2(\vec{P} \cdot \vec{Q}) \]
Simplifying this gives:
\[ 7 = 10 + 2(\vec{P} \cdot \vec{Q}) \implies 2(\vec{P} \cdot \vec{Q}) = 7 - 10 = -3 \implies \vec{P} \cdot \vec{Q} = -\frac{3}{2} \]

### Step 4: Substitute Back into the Expanded Expression
Now substituting back into our expanded expression:
\[ (2\vec{P} + \vec{Q}) \cdot (\vec{P} - \vec{Q}) = 2(1) - 2\left(-\frac{3}{2}\right) + \left(-\frac{3}{2}\right) - 9 \]
Calculating this step by step:
1. \(2(1) = 2\)
2. \(-2\left(-\frac{3}{2}\right) = 3\)
3. \(-\frac{3}{2} = -1.5\)
4. \(-9\)

Combining these:
\[ = 2 + 3 - 1.5 - 9 = 5 - 1.5 - 9 = 3.5 - 9 = -5.5 \]

### Final Answer
Thus, the value of \((2\vec{P} + \vec{Q}) \cdot (\vec{P} - \vec{Q})\) is:
\[ \boxed{-5.5} \]