Question

Help Me! I suck with vectors. Look at the attached pictures for the problem.

Answer 1

Multiplying a vector $x$ by a constant $k$ gives you a new vector whose components are the same as $x$, but scaled by $k$:

$x=\langle x_1,x_2\rangle\implies x=\langle kx_1,kx_2\rangle$

So we have

$u=\langle5,-7\rangle\implies-6u=\langle-30,42\rangle$

$v=\langle-11,3\rangle\implies2v=\langle-22,6\rangle$

Vector addition is just a matter of adding the corresponding components together:

$x=\langle x_1,x_2\rangle,y=\langle y_1,y_2\rangle\implies x+y=\langle x_1+y_1,x_2+y_2\rangle$

Then

$2v-6u=\langle-22,6\rangle+\langle-30,42\rangle=\langle-22-30,6+42\rangle$

$\implies\boxed{2v-6u=\langle-52,48\rangle}$

$\|x\|$ denotes the norm/magnitude of the vector $x$:

$x=\langle x_1,x_2\rangle\implies\|x\|=\sqrt{{x_1}^2+{x_2}^2}$

We have

$\|2v-6u\|=\sqrt{(-52)^2+48^2}=4\sqrt{313}$

$\implies\boxed{\|2v-6u\|\approx70.77}$