Answer:
The required vectors are [tex]u=<-\frac{14}{\sqrt{53}},\frac{4}{\sqrt{53}}>[/tex] and [tex]v=<\frac{14}{\sqrt{53}},-\frac{4}{\sqrt{53}}>[/tex].
Step-by-step explanation:
Given information: P(-4,-2) and Q(3,-4).
We need to find the two vectors parallel to [tex]\overrightarrow {QP}[/tex] with length 2.
If [tex]A(x_1,y_1)[/tex] and [tex]B(x_2,y_2)[/tex], then
[tex]\overrightarrow {AB}=<x_2-x_1,y_2-y_1>[/tex]
[tex]|\overrightarrow {AB}|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
Using the above formula we get
vector QP is,
[tex]\overrightarrow {QP}=<-4-3,-2-(-4)>=<-7,2>[/tex]
Magnitude of vertor QP is,
[tex]|\overrightarrow {QP}|=\sqrt{(-4-3)^2+(-2-(-4))^2}[/tex]
[tex]|\overrightarrow {QP}|=\sqrt{(-7)^2+(2)^2}[/tex]
[tex]|\overrightarrow {QP}|=\sqrt{49+4}[/tex]
[tex]|\overrightarrow {QP}|=\sqrt{53}[/tex]
Using vector is
[tex]\widehat {QP}=\frac{\overline {QP}}{|\overline {QP}|}[/tex]
[tex]\widehat {QP}=\frac{1}{\sqrt{53}}<-7,2>[/tex]
[tex]w=\widehat {QP}=\frac{1}{\sqrt{53}}<-\frac{7}{\sqrt{53}},\frac{2}{\sqrt{53}}>[/tex]
Multiply vector w by 2 to get a parallel vector parallel of QP in same direction.
[tex]u=2w=<-\frac{14}{\sqrt{53}},\frac{4}{\sqrt{53}}>[/tex]
Multiply vector w by -2 to get a parallel vector parallel of QP in opposite direction.
[tex]v=-2w=<\frac{14}{\sqrt{53}},-\frac{4}{\sqrt{53}}>[/tex]
Therefore the required vectors are [tex]u=<-\frac{14}{\sqrt{53}},\frac{4}{\sqrt{53}}>[/tex] and [tex]v=<\frac{14}{\sqrt{53}},-\frac{4}{\sqrt{53}}>[/tex].
