If Vector u=(5,-7) and v=(-11,3), 2v-6u=_____ and ||2v-6u||≈_____

Answer:[tex]2\vec{v}-6\vec{u}=(-52,48) \\ \\ ||2\vec{v}-6\vec{u}||=75.28}[/tex]Step-by-step explanation:In this problem we have two vectors:[tex]\vec{u}=(5,-7) \ and \ \vec{v}=(-11,3)[/tex]So we need to find two things:[tex]2\vec{v}-6\vec{u}[/tex]and:[tex]||2\vec{v}-6\vec{u}||[/tex]FIRST:In this case we have the multiplication of vectors by scalars. A scalar is a simple number, so:[tex]2\vec{v}-6\vec{u} \\ \\ Replace \ \vec{v} \ and \ \vec{u} \ by \ the \ given \ vectors: \\ \\ 2(-11,3)-6(5,-7) \\ \\ Multiply \ each \ component \ by \ the \ corresponding \ scalar:\\ \\ (2\times (-11),2\times 3)+(-6\times 5,-6\times (-7)) \\ \\ (-22,6)+(-30,42) \\ \\ Sum \ of \ vectors: \\ \\ (-22-30,6+42) \\ \\ \therefore \boxed{(-52,48)}[/tex]SECOND:If we name:[tex]\vec{w}=2\vec{v}-6\vec{u}[/tex]Then, [tex]||2\vec{v}-6\vec{u}||[/tex] is the magnitude of the vector [tex]\vec{w}[/tex]. Therefore:[tex]||\vec{w}||=||2\vec{v}-6\vec{u}|| \\ \\ ||\vec{w}||=||(-52,48)|| \\ \\ ||\vec{w}||=\sqrt{(-58)^2+48^2} \\ \\ ||\vec{w}||=\sqrt{3364+2304} \\ \\ ||\vec{w}||=\sqrt{5668} \\ \\ \boxed{||\vec{w}||=75.28}[/tex]