1. Orthogonal vectors

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

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

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

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

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

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

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

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