1. **State the problem:** We need to compute the cross product of vectors \(\mathbf{u} = \begin{bmatrix} 5 \\ -1 \\ -2 \end{bmatrix}\) and \(\mathbf{v} = \begin{bmatrix} -10 \\ 4 \\ 1 \end{bmatrix}\). 2. **Recall the formula for the cross product:** $$\mathbf{u} \times \mathbf{v} = \begin{bmatrix} u_2 v_3 - u_3 v_2 \\ u_3 v_1 - u_1 v_3 \\ u_1 v_2 - u_2 v_1 \end{bmatrix}$$ 3. **Substitute the components:** - \(u_1 = 5\), \(u_2 = -1\), \(u_3 = -2\) - \(v_1 = -10\), \(v_2 = 4\), \(v_3 = 1\) Calculate each component: - First component: \((-1)(1) - (-2)(4) = -1 + 8 = 7\) - Second component: \((-2)(-10) - 5(1) = 20 - 5 = 15\) - Third component: \(5(4) - (-1)(-10) = 20 - 10 = 10\) 4. **Write the cross product vector:** $$\mathbf{u} \times \mathbf{v} = \begin{bmatrix} 7 \\ 15 \\ 10 \end{bmatrix}$$ 5. **Interpretation:** The cross product is not the zero vector; it is \(\begin{bmatrix}7 \\ 15 \\ 10\end{bmatrix}\). This vector is perpendicular to both \(\mathbf{u}\) and \(\mathbf{v}\).