Question

Define the points ​P(negative 4−4​,negative 2−2​) and ​Q(33​,negative 4−4​). Carry out the following calculation. Find two vectors parallel to ModifyingAbove QP with right arrowQP with length 22.

Answer 1

Answer:

The required vectors are $u=<-\frac{14}{\sqrt{53}},\frac{4}{\sqrt{53}}>$ and $v=<\frac{14}{\sqrt{53}},-\frac{4}{\sqrt{53}}>$.

Step-by-step explanation:

Given information: P(-4,-2) and Q(3,-4).

We need to find the two vectors parallel to $\overrightarrow {QP}$ with length 2.

If $A(x_1,y_1)$ and $B(x_2,y_2)$, then

$\overrightarrow {AB}=<x_2-x_1,y_2-y_1>$

$|\overrightarrow {AB}|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Using the above formula we get

vector QP is,

$\overrightarrow {QP}=<-4-3,-2-(-4)>=<-7,2>$

Magnitude of vertor QP is,

$|\overrightarrow {QP}|=\sqrt{(-4-3)^2+(-2-(-4))^2}$

$|\overrightarrow {QP}|=\sqrt{(-7)^2+(2)^2}$

$|\overrightarrow {QP}|=\sqrt{49+4}$

$|\overrightarrow {QP}|=\sqrt{53}$

Using vector is

$\widehat {QP}=\frac{\overline {QP}}{|\overline {QP}|}$

$\widehat {QP}=\frac{1}{\sqrt{53}}<-7,2>$

$w=\widehat {QP}=\frac{1}{\sqrt{53}}<-\frac{7}{\sqrt{53}},\frac{2}{\sqrt{53}}>$

Multiply vector w by 2 to get a parallel vector parallel of QP in same direction.

$u=2w=<-\frac{14}{\sqrt{53}},\frac{4}{\sqrt{53}}>$

Multiply vector w by -2 to get a parallel vector parallel of QP in opposite direction.

$v=-2w=<\frac{14}{\sqrt{53}},-\frac{4}{\sqrt{53}}>$

Therefore the required vectors are $u=<-\frac{14}{\sqrt{53}},\frac{4}{\sqrt{53}}>$ and $v=<\frac{14}{\sqrt{53}},-\frac{4}{\sqrt{53}}>$.