Find the sine of the angle between the vectors `vec(a) =(2 hat(i) - hat(j) + 3 hat(k)) and vec(b) = (hat(i) + 3 hat(j) + 2 hat(k)).`

Explanation

To find the sine of the angle between the vectors \(\vec{a} = (2 \hat{i} - \hat{j} + 3 \hat{k})\) and \(\vec{b} = (\hat{i} + 3 \hat{j} + 2 \hat{k})\), we will use the formula for the sine of the angle between two vectors, which is given by:

\[ \sin \theta = \frac{|\vec{a} \times \vec{b}|}{|\vec{a}| |\vec{b}|} \]

### Step 1: Calculate the magnitudes of vectors \(\vec{a}\) and \(\vec{b}\)

...