To solve for [tex]\(2 \mathbf{v} - 6 \mathbf{u}\)[/tex] and its magnitude, let's start by calculating each component.
Given:
[tex]\[
\mathbf{u} = \langle 5, -7 \rangle
\][/tex]
[tex]\[
\mathbf{v} = \langle -11, 3 \rangle
\][/tex]
First, we multiply vector [tex]\(\mathbf{v}\)[/tex] by 2:
[tex]\[
2 \mathbf{v} = 2 \langle -11, 3 \rangle = \langle 2 \times -11, 2 \times 3 \rangle = \langle -22, 6 \rangle
\][/tex]
Next, we multiply vector [tex]\(\mathbf{u}\)[/tex] by 6:
[tex]\[
6 \mathbf{u} = 6 \langle 5, -7 \rangle = \langle 6 \times 5, 6 \times -7 \rangle = \langle 30, -42 \rangle
\][/tex]
Now, we can calculate [tex]\(2 \mathbf{v} - 6 \mathbf{u}\)[/tex] by subtracting the components of [tex]\(6 \mathbf{u}\)[/tex] from the components of [tex]\(2 \mathbf{v}\)[/tex]:
[tex]\[
2 \mathbf{v} - 6 \mathbf{u} = \langle -22, 6 \rangle - \langle 30, -42 \rangle = \langle -22 - 30, 6 - (-42) \rangle = \langle -52, 48 \rangle
\][/tex]
So, the vector [tex]\(2 \mathbf{v} - 6 \mathbf{u} = \langle -52, 48 \rangle\)[/tex].
Next, we need to calculate the magnitude of this resulting vector [tex]\( \mathbf{w} \)[/tex]:
[tex]\[
\mathbf{w} = \langle -52, 48 \rangle
\][/tex]
The magnitude of [tex]\(\mathbf{w}\)[/tex] is given by:
[tex]\[
\| \mathbf{w} \| = \sqrt{(-52)^2 + 48^2}
\][/tex]
[tex]\[
\| \mathbf{w} \| \approx 70.767
\][/tex]
So, in the dropdown menus, you should select:
[tex]\[
2 \mathbf{v} - 6 \mathbf{u} = \langle -52, 48 \rangle
\][/tex]
[tex]\[
\| 2 \mathbf{v} - 6 \mathbf{u} \| \approx 70.767
\][/tex]
