Question

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

Answer 1

Answer:

$2\vec{v}-6\vec{u}=(-52,48) \\ \\ ||2\vec{v}-6\vec{u}||=75.28}$

Step-by-step explanation:

In this problem we have two vectors:

$\vec{u}=(5,-7) \ and \ \vec{v}=(-11,3)$

So we need to find two things:

$2\vec{v}-6\vec{u}$

and:

$||2\vec{v}-6\vec{u}||$

FIRST:

In this case we have the multiplication of vectors by scalars. A scalar is a simple number, so:

$2\vec{v}-6\vec{u} \\ \\ Replace \ \vec{v} \ and \ \vec{u} \ by \ the \ given \ vectors: \\ \\ 2(-11,3)-6(5,-7) \\ \\ Multiply \ each \ component \ by \ the \ corresponding \ scalar:\\ \\ (2\times (-11),2\times 3)+(-6\times 5,-6\times (-7)) \\ \\ (-22,6)+(-30,42) \\ \\ Sum \ of \ vectors: \\ \\ (-22-30,6+42) \\ \\ \therefore \boxed{(-52,48)}$

SECOND:

If we name:

$\vec{w}=2\vec{v}-6\vec{u}$

Then, $||2\vec{v}-6\vec{u}||$ is the magnitude of the vector $\vec{w}$. Therefore:

$||\vec{w}||=||2\vec{v}-6\vec{u}|| \\ \\ ||\vec{w}||=||(-52,48)|| \\ \\ ||\vec{w}||=\sqrt{(-58)^2+48^2} \\ \\ ||\vec{w}||=\sqrt{3364+2304} \\ \\ ||\vec{w}||=\sqrt{5668} \\ \\ \boxed{||\vec{w}||=75.28}$

Answer 2

Answer:

$2v-6u=(-52,48)$

$||2\vec{v}-6\vec{u}||=75.28$

Step-by-step explanation:

Given : If Vector u=(5,-7) and v=(-11,3)

To find : The value of 2v-6u and ||2v-6u||?

Solution :  

We have given,

$\vec{u}=(5,-7)$ and $\vec{v}=(-11,3)$

Substitute in $2v-6u$

$= 2(-11,3)-6(5,-7)$

$= (2\times (-11),2\times 3)+(-6\times 5,-6\times (-7))$

$=(-22,6)+(-30,42)$

Adding the two vectors,

$=(-22-30,6+42)$

$=(-52,48)$

So, $2v-6u=(-52,48)$

Now, we find the value of $||2\vec{v}-6\vec{u}||$

Substitute the value,

$||2\vec{v}-6\vec{u}||=||(-52,48)|| \\ \\ ||||2\vec{v}-6\vec{u}||=\sqrt{(-58)^2+48^2} \\ \\ ||||2\vec{v}-6\vec{u}||=\sqrt{3364+2304} \\ \\ ||2\vec{v}-6\vec{u}||=\sqrt{5668} \\ \\||2\vec{v}-6\vec{u}||=75.28$

So, $||2\vec{v}-6\vec{u}||=75.28$