Let be orthogonal unit vectors.
- Prove that the vectors form an orthonormal basis for .
- Prove that the vector has unit length.
- Show the geometric relation between and obtain the relations
- Prove the relations in part (c) algebraically.
- Proof. By Theorem 13.13 (page 484 of Apostol) we know that if and are independent then so is the set . We also know by Theorem 13.12 (page 483) that is orthogonal to both and . Since this is a set of three independent vectors in , it is a basis. Then, if each have length 1 and are orthogonal we have
Hence, has length 1 as well. Thus, is form an orthonormal basis
- Proof. From part (a) we know that and each have length 1 and are orthogonal. Thus,
- Proof. Since are orthogonal, we know every vector orthogonal to two of them is a scalar multiple of the third. Thus, is a scalar multiple of . Further, since all have length 1
If we adopt a right hand coordinate system then . So, . Similarly,