Thread: Find vectors that satisfy the following conditions

Find vectors and \in \mathbb{R}^3 that simultaneously satisfy all of the following:

-
- is parallel to (2,1,5)
- 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:





and



also that



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

Thanks.

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).