find the cosine of the angle between the vectors `vec a= 3` `hat i`+2 `hat k` and
`vec b = 2 hat i` -2 `hat j` + 4 `hat k`

Explanation

To find the cosine of the angle between the vectors \(\vec{a} = 3\hat{i} + 2\hat{k}\) and \(\vec{b} = 2\hat{i} - 2\hat{j} + 4\hat{k}\), we will follow these steps:

### Step 1: Calculate the dot product \(\vec{a} \cdot \vec{b}\)

The dot product of two vectors \(\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\) and \(\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}\) is given by:

\[
\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3
...