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- 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 = (x

We then have (x, u) = x

(x, v) = 7x

and (x, w) = x

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.

We can let x = (x

_{1}, x_{2}, x_{3}, x_{4}) be a vector orthogonal to u, v, and w.We then have (x, u) = x

_{1}- x_{2}+ 6 x_{3}= 0,(x, v) = 7x

_{1}+ x_{2}+ x_{4}= 0,and (x, w) = x

_{1}+ 4 x_{3}+ x_{4}= 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.