Find two vectors v1 and v2 whose sum is <3,-1> , where v1 is parallel to <-5,1> while v2 is perpendicular to <-5,1>

Question

Find two vectors v1 and v2 whose sum is <3,-1> , where v1 is parallel to <-5,1> while v2 is perpendicular to <-5,1>. Find v1 and v2.

Answer 1

Parallel vectors will be multiples of each other.

Perpendicular vectors will have a dot product of 0.

Let v1 = <a,b> and v2 = <c,d>

v1 || <-5,1> means v1 = N <-5,1> = <a,b>

So a = -5N and b=N (keep this handy for later)

v2 • <-5,1> = 0 (because they are perpendicular)

So <c,d> • <-5,1> = -5c + d = 0 or d = 5c (keep this handy for later)

v1 + v2 = <a,b> + <c,d> = <3, -1>

So a + c = 3 and b + d = -1 (keep this handy - putting all together now)

Substitute -5N for a. Substitute N for b. Substitute 5c for d.

This gives us a system of equations with two equations and 2 unknowns:

-5N + c = 3

N + 5c = -1

Multiply everything in the bottom equation by 5

-5N + c = 3

5N + 25c = -5

Now add vertically, eliminating the variable N and finding

26c = -2

or

c = -1/13

Now we work our way backwards.

-5N + c = -5N + (-1/13) = 3

-5N = 40/13

(-1/5) (-5N) = (40/13)(-1/5)

and N = -8/13 and b = N so b = -8/13

d = 5c so d = -5/13

a = -5N, so a = 40/13

So the two vectors are:

v1 = < 40/13, -8/13 >

v2 = < -1/13, -5/13 >