If `vec(a)=2hat(i)+hat(j)+3hat(k),vec(b)=-hat(i)+2hat(j)+hat(k)`, and `vec(c)=-3hat(i)+hat(j)+2hat(k)`, find `[hat(a)hat(b)hat(c)]`.

Explanation

To find the scalar triple product \([ \vec{a} \, \vec{b} \, \vec{c} ]\) of the vectors \(\vec{a} = 2\hat{i} + \hat{j} + 3\hat{k}\), \(\vec{b} = -\hat{i} + 2\hat{j} + \hat{k}\), and \(\vec{c} = -3\hat{i} + \hat{j} + 2\hat{k}\), we can use the determinant of a matrix formed by these vectors.

### Step 1: Set up the determinant
We will create a \(3 \times 3\) matrix where the first row contains the coefficients of \(\vec{a}\), the second row contains the coefficients of \(\vec{b}\), and the third row contains the coefficients of \(\vec{c}\).

\[
\begin{vmatrix}
2 & 1 & 3 \\
...