Finding Perpendicular Vectors with Sum [6,8]

[PiRsq]

Find 2 vectors u and v such that they are perpendicular one of the vector is twice the magnitude of the other. And the sum vector of u and v is [6,8]

I did:

Let u=[a,b]
Let v=[c,d]
Let |u|=2|v

u.v=ac+bd=0

|u+v|=|u|^2 + |v|^2

But |u|=2|v|

|u+v|=5|v|^2

5|v|^2=100

|v|^2=20

Im stuck after this

 

[Hurkyl]

You forgot to write ^2 on |u+v|

Anyways, can you write |v|^2 in terms of a, b, c, and d? Do you know what |u|^2 is? Have you used the fact that u + v = [6, 8] yet?

 

[PiRsq]

So

|u|^2+|v|^2=|u+v|^2

But since |u|^2=4|v|^2

5|v|^2=36+64
|v|^2=20

now where can I go?

 

[Hurkyl]

Can you write |v|^2 in terms of a, b, c, and d? Do you know what |u|^2 is? Have you used the fact that u + v = [6, 8] yet?

 

[PiRsq]

|v|^2 = c^2+d^2

|v|^2 = 1/2 (a^2 + b^2)


1/2(a^2+ b^2)= c^2 + d^2

Is that what you mean?

 

[Hurkyl]

Can you think of anything better you can do with those first two equatinos?

 

[PiRsq]

Since |v|^2=2|u|^2,

|v|^2 = c^2+d^2

|u|^2 = 1/2 (c^2 + d^2)


|u|^2 + |v|^2 = 100

3/2 (c^2 + d^2) = 100

I don't know where I am headed

 

[Hurkyl]

what formulas do you have involving |v|^2?

 

[PiRsq]

Projection of u on v

u on v = [u.v/|v|^2] |v|

 

[Hurkyl]

I guess I should have asked this first off...

You know that your goal is to find 4 equations involving only a, b, c, d which you can solve, right?

 

[PiRsq]

4 equations?? I didnt know that...I've been focusing on finding a and b

 

[Hurkyl]

Well, you have 4 unknowns; a, b, c, and d. In general, you need 4 equations to solve for all 4 of them.

Sometimes you can get lucky and you can find two equations that involve only a and b, but in general that won't happen (and I'm pretty sure it doesn't here)...


You've already found one good equation:

ac + bd = 0


You just need 3 more! You can get 2 more equations out of what you've told me about |v|^2...


Oh, BTW, if |u| = 2|v|, then |u|^2 = 4|v|^2

 

[PiRsq]

Ill try this and post a little later, thanks man