# Finding Orthogonal Unit Vector to 3 Vectors

• ver_mathstats

#### 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.

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\|}##.