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# Prove some facts about orthogonal unit vectors

Let be orthogonal unit vectors.

1. Prove that the vectors form an orthonormal basis for .
2. Prove that the vector has unit length.
3. Show the geometric relation between and obtain the relations 4. Prove the relations in part (c) algebraically.

1. 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 2. Proof. From part (a) we know that and each have length 1 and are orthogonal. Thus, 3. Incomplete.
4. 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, 