1. ## Orthogonal vectors

Let $\mathbf{a,b,c}$ be three vectors in $\mathbb{R}^3$ which satisfy the relations $\mathbf{b=c \times a}\ and\ \mathbf{c=a \times b}$

1) Show that $\mathbf{a, b\ and\ c}$ are a set of mutually orthogonal vectors

2) Show that $\mathbf{b\ and\ c}$ are of equal length and that if $\mathbf{b \neq 0}$, then $\mathbf{a}$ is a unit vector.

2. Originally Posted by acevipa
Let $\mathbf{a,b,c}$ be three vectors in $\mathbb{R}^3$ which satisfy the relations $\mathbf{b=c \times a}\ and\ \mathbf{c=a \times b}$

1) Show that $\mathbf{a, b\ and\ c}$ are a set of mutually orthogonal vectors

2) Show that $\mathbf{b\ and\ c}$ are of equal length and that if $\mathbf{b \neq 0}$, then $\mathbf{a}$ is a unit vector.
This is pretty straight forward isn't it? $\mathbf{b= c\times a}$ is, by definition of "cross product", orthogonal to both $\mathbf{a}$ and $mathbf{c}$ and $\mathbf{c= a\times b}$ is orthogonal to a.

Further, since $\mathbf{a}$ is orthogonal to $\mathbf{c}$, $|\mathbf{b}|= |\mathbf{a}\times\mathbf{c}|= |\mathbf{a}||\mathbf{c}\sin(\pi/2)$ so that

$1= (|\mathbf{a}|)(1)(1)$.