Find vectors that satisfy the following conditions

Find vectors $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$ \in \mathbb{R}^3 that simultaneously satisfy all of the following:

- $\displaystyle \vec{u}-\vec{v} = (3,4,10)$

- $\displaystyle \vec{u}$ is parallel to (2,1,5)

- $\displaystyle \vec{v}$ is perpendicular to (2,1,5)

I understand what each condition means in terms of how it affects the vectors I am trying to find; however, I can't seem to relate them enough to derive the actual vectors. I know that:

$\displaystyle u_1 - v_1 = 3$

$\displaystyle u_2 - v_2 = 4$

$\displaystyle u_3 - v_3 = 10$

and

$\displaystyle \vec{u} = (2m,1m,5m)$

also that

$\displaystyle \vec{v}.(2,1,5) = 0$

How do I go about finding the two vectors that satisfy these conditions other than trial and error?

Thanks.

Re: Find vectors that satisfy the following conditions

from u1 = 2m, u2 = m, u3 = 5m, and u1-v1 = 3, u2-v2 = 4, u3-v3 = 10 we have:

v1 = u1 - 3 = 2m - 3

v2 = u2 - 4 = m - 4

v3 = u3 - 10 = 5m - 10.

from 2v1 + v2 + 5v3 = 0, we have:

2(2m-3) + m-4 + 5(5m-10) = 0, so

4m - 6 + m - 4 + 25m - 50 = 0

30m = 60

m = 2.

hence u = (4,2,10) and v = (1,-2,0).