For the vectors \( \mathbf{u}=\langle-3,2\rangle \) and \( \mathbf{v}=\langle-1,5\rangle \), calculate proj \( _{\mathbf{v}} \mathbf{u} \) and scal \( _{\mathbf{v}} \mathbf{u} \). Points: 0 of 1 proj \( _{\mathbf{v}} \mathbf{u}=\left\langle-\frac{1}{2}, \frac{5}{2}\right\rangle \) scal \( _{\mathbf{v}} \mathbf{u}=\square \) (Type an exact answer, using radicals as needed.)

To calculate the projection of \(\mathbf{u}\) onto \(\mathbf{v}\) and the scalar component, we can use the following formulas: 1. The projection of \(\mathbf{u}\) onto \(\mathbf{v}\): \[ \text{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v} \] 2. The scalar component of \(\mathbf{u}\) in the direction of \(\mathbf{v}\): \[ \text{scal}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|} \] First, we calculate \(\mathbf{u} \cdot \mathbf{v}\): \[ \mathbf{u} \cdot \mathbf{v} = (-3)(-1) + (2)(5) = 3 + 10 = 13 \] Next, we find \(\mathbf{v} \cdot \mathbf{v}\): \[ \mathbf{v} \cdot \mathbf{v} = (-1)(-1) + (5)(5) = 1 + 25 = 26 \] Now we can find the projection: \[ \text{proj}_{\mathbf{v}} \mathbf{u} = \frac{13}{26} \mathbf{v} = \frac{1}{2} \langle -1, 5 \rangle = \langle -\frac{1}{2}, \frac{5}{2} \rangle \] Now let's calculate \(\|\mathbf{v}\|\) to find the scalar: \[ \|\mathbf{v}\| = \sqrt{(-1)^2 + 5^2} = \sqrt{1 + 25} = \sqrt{26} \] Thus, we can find the scalar component: \[ \text{scal}_{\mathbf{v}} \mathbf{u} = \frac{13}{\sqrt{26}} = \frac{13 \sqrt{26}}{26} = \frac{\sqrt{26}}{2} \] So the final answers are: \[ \text{proj}_{\mathbf{v}} \mathbf{u} = \left\langle -\frac{1}{2}, \frac{5}{2} \right\rangle \] \[ \text{scal}_{\mathbf{v}} \mathbf{u} = \frac{\sqrt{26}}{2} \]