Find the area of the parallelogram whose adjacent sides are represented by the vectors `(3 hat (i)+hat(j)-2 hat(k)) and (hat (i)-3 hat(j)+4 hat (k)).`

Explanation

To find the area of the parallelogram formed by the vectors \( \mathbf{a} = 3\hat{i} + \hat{j} - 2\hat{k} \) and \( \mathbf{b} = \hat{i} - 3\hat{j} + 4\hat{k} \), we can use the formula for the area of a parallelogram defined by two vectors, which is given by the magnitude of their cross product:

\[ \text{Area} = |\mathbf{a} \times \mathbf{b}| \]

### Step 1: Set up the cross product
To find the cross product \( \mathbf{a} \times \mathbf{b} \), we can use the determinant of a matrix formed by the unit vectors \( \hat{i}, \hat{j}, \hat{k} \) and the components of the vectors \( \mathbf{a} \) and \( \mathbf{b} \):
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