Linear Algebra: find a unique vector v given two other vectors related to v

[greenclipboar]

Homework Statement

Let u=[1,1,-1] and w=[2,4,2].
Find the unique vector v=[v1,v2,v3] such that:
v.u=1
v is orthogonal to w
|v|=sqrt(3)
v2 > 0

Homework Equations

v.u=|v||u|cos(theta)

The Attempt at a Solution

I've just been blindly solving different aspects of this, but have no idea how to put it all together.

w.v=0

u.w=4

v.u=v1+v2-v3=1

|v|=sqrt(3) = sqrt(v1^2+v2^2+v3^2) therefore v1^2+v2^2+v3^2 = 3

cos(theta) between u and v is 1/3

cos(theta) between w and v is 1/sqrt(6)

 

[greenclipboar]

Never mind, I figured it out. You solve for v1 and v3 using the above relations, then plus both of those into v1^2+v2^2+v3^2 = 3 to get a quadratic only using v2.
The solve for v2 (only the positive answer) which you sub into any equation to find v1 and v3.