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

The required vectors are
`u=<-(14)/(√(53)),(4)/(√(53))>` and
`v=<(14)/(√(53)),-(4)/(√(53))>`.

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}|=√((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}|=√((-4-3)^2+(-2-(-4))^2)`

`|\overrightarrow {QP}|=√((-7)^2+(2)^2)`

`|\overrightarrow {QP}|=√(49+4)`

`|\overrightarrow {QP}|=√(53)`

Using vector is

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

`\widehat {QP}=(1)/(√(53))<-7,2>`

`w=\widehat {QP}=(1)/(√(53))<-(7)/(√(53)),(2)/(√(53))>`

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

`u=2w=<-(14)/(√(53)),(4)/(√(53))>`

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

`v=-2w=<(14)/(√(53)),-(4)/(√(53))>`

Therefore the required vectors are
`u=<-(14)/(√(53)),(4)/(√(53))>` and
`v=<(14)/(√(53)),-(4)/(√(53))>`.