proof orthogonal nonzero vectors

Question

Show that if x,y,z are orthogonal nonzero vectors in R³ , then form a basis for R³

Answer 1

Writing a proof for a homework exercise depends typically on what tools you are allowed to use. If we are allowed to use the fact that if the determinant of a 3 x 3 matrix is nonzero, its columns are linearly independent, then we can just compute the determinant and show it is nonzero. This would show that the vectors are linearly independent.

Let the vectors be v1,v2, and v3 and A = [v1,v2,v3] be the matrix. Then,

det(A) = det(A^T) = v₁•(v₂Xv₃) (this is triple scalar product).

Now, because v₂Xv₃ is orthogonal to both v₂ and v₃, we know it must be parallel or antiparallel to v₁. Therefore, v₁•(v₂Xv₃) ≠ 0.

Thus, the three vectors are linearly independent. Therefore, they must form a basis for R³. QED