Finding Orthogonal Unit Vector to 3 Vectors

[ver_mathstats]

Homework Statement

Let R4 have the Euclidean inner product. Find a unit vector with a positive first component that is orthogonal to all three of the following vectors.

Relevant Equations

u = (1,-1,6,0) v = (7,1,0,1) w = (1,0,4,1)

I'm having difficulties solving this. For finding a unit vector that is orthogonal to two unit vectors I understand we use the cross product and such. However, I am confused about how to approach this problem as it has a third vector.

We can let x = (x₁, x₂, x₃, x₄) be a vector orthogonal to u, v, and w.

We then have (x, u) = x₁ - x₂ + 6 x₃ = 0,
(x, v) = 7x₁ + x₂ + x₄ = 0,
and (x, w) = x₁ + 4 x₃ + x₄ = 0.

I am just a little bit confused on where to go from here, would I be setting up a matrix? Any help would be appreciated, thanks.

 

[fresh_42]

You have set up all conditions for orthogonality. Now $x_1>0$ and $\|x\|=1$ are yet to guarantee. I would simply choose $x_1=1$, solve the equation system and finally calculate $\dfrac{x}{\|x\|}$.