Prove some facts about orthogonal unit vectors

Let $A,B \in \mathbb{R}^3$ be orthogonal unit vectors.

Prove that the vectors $A, \ B, A \times B$ form an orthonormal basis for $\mathbb{R}^3$ .
Prove that the vector $C = (A \times B) \times A$ has unit length.
Show the geometric relation between $A, \ B, A \times B$ and obtain the relations
$(A \times B) \times A = B, \qquad (A \times B)\times B = -A.$
Prove the relations in part (c) algebraically.

Proof. By Theorem 13.13 (page 484 of Apostol) we know that if $A$ and $B$ are independent then so is the set $\{ A, B, A \times B \}$ . We also know by Theorem 13.12 (page 483) that $A \times B$ is orthogonal to both $A$ and $B$ . Since this is a set of three independent vectors in $\mathbb{R}^3$ , it is a basis. Then, if $A,B$ each have length 1 and are orthogonal we have
$\lVert A \times B \rVert = \lVert A \rVert \lVert B \rVert \sin \theta = (1)(1)\left( \sin \frac{\pi}{2} \right) = 1.$

Hence, $A \times B$ has length 1 as well. Thus, $\{ A, B, A \times B \}$ is form an orthonormal basis $. \qquad \blacksquare$
Proof. From part (a) we know that $(A \times B)$ and $A$ each have length 1 and are orthogonal. Thus,
$\lVert C \rVert^2 = \lVert A \times B \rVert^2 \lVert A \rVert^2 \sin \frac{\pi}{2} = (1)(1)(1) = 1. \qquad \blacksquare$
Incomplete.
Proof. Since $A,B,A \times B$ are orthogonal, we know every vector orthogonal to two of them is a scalar multiple of the third. Thus, $(A \times B) \times A$ is a scalar multiple of $B$ . Further, since $A, B, A \times B$ all have length 1
$\lVert (A \times B) \times A \rVert = \lVert cB \rVert \quad \implies \quad c = \pm 1.$

If we adopt a right hand coordinate system then $c = 1$ . So, $(A \times B) \times A = B$ . Similarly, $(A \times B) \times B = -A. \qquad \blacksquare$