Given \( \mathbf{u} \) and \( \mathbf{v} \), find \( \operatorname{comp}_{\mathbf{v}} \mathbf{u} \) \[ \begin{array}{l}\mathbf{u}=[-8,5,-4] \\ \mathbf{v}=[7,-5,-6] \\ \operatorname{comp}_{\mathbf{v}} \mathbf{u}=\operatorname{Ex:~} 1.234\end{array} \]

To find the component of vector \(\mathbf{u}\) in the direction of vector \(\mathbf{v}\), we use the formula: \[ \operatorname{comp}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|} \] First, we calculate the dot product \(\mathbf{u} \cdot \mathbf{v}\): \[ \mathbf{u} \cdot \mathbf{v} = (-8)(7) + (5)(-5) + (-4)(-6) = -56 - 25 + 24 = -57 \] Next, we need the magnitude of vector \(\mathbf{v}\): \[ \|\mathbf{v}\| = \sqrt{(7)^2 + (-5)^2 + (-6)^2} = \sqrt{49 + 25 + 36} = \sqrt{110} \] Now, we can substitute these values back into our formula for the component: \[ \operatorname{comp}_{\mathbf{v}} \mathbf{u} = \frac{-57}{\sqrt{110}} \] Calculating the numerical value: \[ \sqrt{110} \approx 10.4881 \] Thus, \[ \operatorname{comp}_{\mathbf{v}} \mathbf{u} \approx \frac{-57}{10.4881} \approx -5.44 \] Therefore, the component of \(\mathbf{u}\) in the direction of \(\mathbf{v}\) is approximately: \[ \operatorname{comp}_{\mathbf{v}} \mathbf{u} \approx -5.44 \]