Find two vectors v1 and v2 whose sum is -4-5 , where v1 is parallel to 0-5 while v2 is perpendicular to 0-5

Question

Find V1 and V2

Answer 1

Assuming -4-5 represents -4i -5j, 0-5 represents -5j and that the vectors are in the same plane.

 

v₁ + v₂ = -4i -5j

 

If v₁ is parallel to -5j, then v₁ = aj

 

If v₂ is perpendicular to -5j, then v₂ = bi

 

Then v₁ + v₂ =  bi + aj = -4i -5j

 

Then a = -5 and b = -4, so that v₁ = -5j and v₂ = -4i

Answer 2

let's see

V1=<x1,y1>    V2=<x2,y2>

x1+x2=-4

y1+y2=-5

 

V2 is perpendicular to <0,5>

so V2.<0,5>=0 ==> 0.x2+5y2=0 ---> y2=0

from equation in forth line y1=-5

so V2=<x2,0>       V1=<x1,-5>

now if V2 is perpendicular to <0,5> and V1 is parallel to <0,5>

that  must mean that V2 is also perpendicular to V1

V1.V2=<x1,-5><x2,0>    -->  x1.x2=0

x1=-4-x2

(-4-x2)x2=0

x2=0 or x2=-4x2=0 would result just in 0 vector it is a boring case!

x2=-4 ---> x1=0

so V2=<-4,0> and V1=<0,-5>

 

check!

V1+V2 does give <-4,-5>

V1 is parallel to <0.5>

and V2 is perpendicular to <0,5>

and also V2 is perpendicular to <0,-5>