Let's work through the provided problem in a detailed, step-by-step manner:
1. Given vectors:
[tex]\[ u = \langle 5, -7 \rangle \][/tex]
[tex]\[ v = \langle -11, 3 \rangle \][/tex]
2. We need to compute [tex]\(2v - 6u\)[/tex].
3. Calculating [tex]\(2v\)[/tex]:
[tex]\[ 2v = 2 \cdot \langle -11, 3 \rangle = \langle 2 \cdot (-11), 2 \cdot 3 \rangle = \langle -22, 6 \rangle \][/tex]
4. Calculating [tex]\(6u\)[/tex]:
[tex]\[ 6u = 6 \cdot \langle 5, -7 \rangle = \langle 6 \cdot 5, 6 \cdot (-7) \rangle = \langle 30, -42 \rangle \][/tex]
5. Subtracting [tex]\(6u\)[/tex] from [tex]\(2v\)[/tex] to get [tex]\(2v - 6u\)[/tex]:
[tex]\[ 2v - 6u = \langle -22, 6 \rangle - \langle 30, -42 \rangle \][/tex]
[tex]\[ = \langle -22 - 30, 6 - (-42) \rangle \][/tex]
[tex]\[ = \langle -52, 48 \rangle \][/tex]
So, [tex]\(2v - 6u = \langle -52, 48 \rangle \)[/tex].
6. Next, we need to find the Euclidean norm of the resultant vector [tex]\( \langle -52, 48 \rangle \)[/tex].
7. The Euclidean norm (magnitude) of a vector [tex]\(\langle a, b \rangle\)[/tex] is given by:
[tex]\[ \| \langle -52, 48 \rangle \| = \sqrt{(-52)^2 + 48^2} \][/tex]
8. Calculating:
[tex]\[ = \sqrt{2704 + 2304} \][/tex]
[tex]\[ = \sqrt{5008} \][/tex]
[tex]\[ \approx 70.76722405181653 \][/tex]
Hence, the answers are:
[tex]\[ 2v - 6u = \langle -52, 48 \rangle \][/tex]
[tex]\[ \|2v - 6u \| \approx 70.76722405181653 \][/tex]
Therefore, you should select the following in the drop-down menus:
[tex]\[2 v - 6 u = \langle -52, 48 \rangle \quad E\][/tex]
[tex]\[ \|2 v - 6u \| \approx \quad 70.76722405181653 \, \text{(or the closest rounded value given in the choices)}. \][/tex]
