Multiplying a vector [tex]x[/tex] by a constant [tex]k[/tex] gives you a new vector whose components are the same as [tex]x[/tex], but scaled by [tex]k[/tex]:
[tex]x=\langle x_1,x_2\rangle\implies x=\langle kx_1,kx_2\rangle[/tex]
So we have
[tex]u=\langle5,-7\rangle\implies-6u=\langle-30,42\rangle[/tex]
[tex]v=\langle-11,3\rangle\implies2v=\langle-22,6\rangle[/tex]
Vector addition is just a matter of adding the corresponding components together:
[tex]x=\langle x_1,x_2\rangle,y=\langle y_1,y_2\rangle\implies x+y=\langle x_1+y_1,x_2+y_2\rangle[/tex]
Then
[tex]2v-6u=\langle-22,6\rangle+\langle-30,42\rangle=\langle-22-30,6+42\rangle[/tex]
[tex]\implies\boxed{2v-6u=\langle-52,48\rangle}[/tex]
[tex]\|x\|[/tex] denotes the norm/magnitude of the vector [tex]x[/tex]:
[tex]x=\langle x_1,x_2\rangle\implies\|x\|=\sqrt{{x_1}^2+{x_2}^2}[/tex]
We have
[tex]\|2v-6u\|=\sqrt{(-52)^2+48^2}=4\sqrt{313}[/tex]
[tex]\implies\boxed{\|2v-6u\|\approx70.77}[/tex]
