Evaluate : `(3vec(a)-5vec(b)).(2vec(a)+7vec(b))`.

Answer

To evaluate the expression \((3\vec{a} - 5\vec{b}) \cdot (2\vec{a} + 7\vec{b})\), we will use the distributive property of the dot product. Here’s the step-by-step solution:

### Step 1: Expand the expression using the distributive property
We can expand the dot product as follows:

\[ (3\vec{a} - 5\vec{b}) \cdot (2\vec{a} + 7\vec{b}) = 3\vec{a} \cdot 2\vec{a} + 3\vec{a} \cdot 7\vec{b} - 5\vec{b} \cdot 2\vec{a} - 5\vec{b} \cdot 7\vec{b} \]

### Step 2: Calculate each term
Now, we calculate each term separately:

1. **First term**:
\[ 3\vec{a} \cdot 2\vec{a} = 6(\vec{a} \cdot \vec{a}) = 6|\vec{a}|^2 \]

2. **Second term**:
\[ 3\vec{a} \cdot 7\vec{b} = 21(\vec{a} \cdot \vec{b}) \]

3. **Third term**:
\[ -5\vec{b} \cdot 2\vec{a} = -10(\vec{b} \cdot \vec{a}) = -10(\vec{a} \cdot \vec{b}) \]

4. **Fourth term**:
\[ -5\vec{b} \cdot 7\vec{b} = -35(\vec{b} \cdot \vec{b}) = -35|\vec{b}|^2 \]

### Step 3: Combine the results
Now, we combine all the terms:

\[ 6|\vec{a}|^2 + 21(\vec{a} \cdot \vec{b}) - 10(\vec{a} \cdot \vec{b}) - 35|\vec{b}|^2 \]

This simplifies to:

\[ 6|\vec{a}|^2 + (21 - 10)(\vec{a} \cdot \vec{b}) - 35|\vec{b}|^2 \]

\[ = 6|\vec{a}|^2 + 11(\vec{a} \cdot \vec{b}) - 35|\vec{b}|^2 \]

### Final Result
Thus, the final result of the expression \((3\vec{a} - 5\vec{b}) \cdot (2\vec{a} + 7\vec{b})\) is:

\[ 6|\vec{a}|^2 + 11(\vec{a} \cdot \vec{b}) - 35|\vec{b}|^2 \]

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