Find the area of the parallelogram spanned by the following vectors. \[ \begin{array}{l}\mathbf{u}=[0,0,3] \\ \mathbf{v}=[8,6,4] \\ \text { Area of parallelogram }=\text { Ex: } 1.234\end{array} \]

To find the area of the parallelogram spanned by the vectors \(\mathbf{u}\) and \(\mathbf{v}\), we can use the cross product of these two vectors. 1. First, compute the cross product \(\mathbf{u} \times \mathbf{v}\): \[ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & 0 & 3 \\ 8 & 6 & 4 \end{vmatrix} = \mathbf{i}(0 \cdot 4 - 3 \cdot 6) - \mathbf{j}(0 \cdot 4 - 3 \cdot 8) + \mathbf{k}(0 \cdot 6 - 0 \cdot 8) \] \[ = \mathbf{i}(0 - 18) - \mathbf{j}(0 - 24) + \mathbf{k}(0 - 0) = -18\mathbf{i} + 24\mathbf{j} \] 2. Next, find the magnitude of the cross product: \[ \|\mathbf{u} \times \mathbf{v}\| = \sqrt{(-18)^2 + (24)^2} = \sqrt{324 + 576} = \sqrt{900} = 30 \] Therefore, the area of the parallelogram is \(30\). Area of parallelogram = 30