91Ó°ÊÓ

Question: Compute the cross product of the two vectors \(\mathbf{u} = (2, 3, 4)\) and \(\mathbf{v} = (1, -1, 2)\) using the determinant approach. Answer: Following the steps outlined in the solution, the cross product \(\mathbf{u} \times \mathbf{v}\) can be computed as follows: 1. Write the vectors in component form: \(\mathbf{u} = (2, 3, 4)\) and \(\mathbf{v} = (1, -1, 2)\). 2. Set up the 3x3 matrix: $$ \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 3 & 4 \\ 1 & -1 & 2 \\ \end{vmatrix} $$ 3. Compute the determinant: $$ \mathbf{u} \times \mathbf{v} = \mathbf{i} \begin{vmatrix} 3 & 4 \\ -1 & 2 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 2 & 4 \\ 1 & 2 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 2 & 3 \\ 1 & -1 \end{vmatrix} $$ $$ \mathbf{u} \times \mathbf{v} = \mathbf{i} (3 \cdot 2 - 4 \cdot (-1)) - \mathbf{j} (2 \cdot 2 - 4 \cdot 1) + \mathbf{k} (2 \cdot (-1) - 3 \cdot 1) $$ 4. Write the cross product in component form: $$ \mathbf{u} \times \mathbf{v} = (10, 0, -5) $$ Thus, the cross product of \(\mathbf{u}\) and \(\mathbf{v}\) is \(\mathbf{u} \times \mathbf{v} = (10, 0, -5)\).