Given \( \mathbf{u} \) and \( \mathbf{v} \), find \( \operatorname{comp}_{\mathbf{v}} \mathbf{u} \) \( \mathbf{u}=[-9,-7] \) \( \mathbf{v}=[-1,-8] \) \( \operatorname{comp}_{\mathbf{v}} \mathbf{u}= \) Ex: 1.234

To find the component of vector \( \mathbf{u} \) in the direction of vector \( \mathbf{v} \), we can use the formula: \[ \operatorname{comp}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|} \] First, we calculate the dot product \( \mathbf{u} \cdot \mathbf{v} \): \[ \mathbf{u} \cdot \mathbf{v} = (-9)(-1) + (-7)(-8) = 9 + 56 = 65 \] Next, we calculate the magnitude of \( \mathbf{v} \): \[ |\mathbf{v}| = \sqrt{(-1)^2 + (-8)^2} = \sqrt{1 + 64} = \sqrt{65} \] Now, we can plug in the values into the component formula: \[ \operatorname{comp}_{\mathbf{v}} \mathbf{u} = \frac{65}{\sqrt{65}} = \sqrt{65} \] So, the solution is: \[ \operatorname{comp}_{\mathbf{v}} \mathbf{u} = \sqrt{65} \approx 8.062 \]