Consider R4 with the Euclidean inner product.

(a) Find a unit vector that is orthogonal to the vectors u = (2, 1, −4, , 0), v = (−1, −1, 2, 2), and w = (3, 2, 4, 5).

(b) Let W be the subspace of R4 spanned by the vectors u, v, and w. Find a basis for the orthogonal complement of W.

For part a, how do I find the orthogonal unit vector for u,v and w? I've only ever found orthogonal unit vectors for 2 vectors, not 3.

For part b, how do you find the orthogonal complement of a basis?

Any help would be appreciated.

  • $\begingroup$ These two problems are redundant (if the three vectors are linearly independent, that is). Once you’ve solved (a), you’ve already got a solution for (b), while solving (b) gives you a vector that at most needs to be normalized to solve (a). $\endgroup$
    – amd
    Dec 9, 2015 at 3:50

1 Answer 1


For a, I would make a 4x4 matrix with the last row (or column) being the individual unit vectors that span your 4-space, then solve it by taking the determinant. Just like you do in Cartesian space but with one extra dimension. Then normalize it to make it a unit vector.

  • $\begingroup$ This will give you a vector that’s linearly independent of the others, but how does it guarantee that you’ll end up with a vector that’s orthogonal to them? $\endgroup$
    – amd
    Dec 9, 2015 at 3:58
  • $\begingroup$ This determinant gives you the cross product. The cross product is used to find mutually orthogonal vectors. $\endgroup$
    – SteveO
    Dec 9, 2015 at 4:00
  • $\begingroup$ Ah, yes. Very nice. $\endgroup$
    – amd
    Dec 9, 2015 at 4:04

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