Find `( vec a+3 vec b).(2 vec a- vec b)`, If `vec a= hat i+ hat j+2 hat k` and `vec b=3 hat i+2 hat j- hat k`

Explanation

To solve the problem of finding \((\vec{a} + 3\vec{b}) \cdot (2\vec{a} - \vec{b})\) given \(\vec{a} = \hat{i} + \hat{j} + 2\hat{k}\) and \(\vec{b} = 3\hat{i} + 2\hat{j} - \hat{k}\), we will follow these steps:

### Step 1: Calculate \(\vec{a} + 3\vec{b}\)

1. Start with the expression \(\vec{a} + 3\vec{b}\).
2. Substitute the values of \(\vec{a}\) and \(\vec{b}\):
\[
\vec{a} = \hat{i} + \hat{j} + 2\hat{k}
...