If ` vec a ,\ vec b ,\ vec c` are three unit vecrtors such that ` vec axx vec b= vec c ,\ vec bxx vec c= vec a ,\ vec cxx vec a= vec b` Show that ` vec a ,\ vec b ,\ vec c` form an orthonormal right handed triad of unit vectors.

Explanation

` vec axx vec b= vec c ,\ vec bxx vec c= vec a ,\ vec cxx vec a= vec b`
` (vec a xx vec b)xxvec a = vec c xx vec a `
`=>(vec a * vec a )vec b -(vec a * vec b )veca = vec c xx vec a `
`=>vec b -(vec c * vec b )vec a = vec b`
`=>(vec c * vec b )vec a=0`
`therefore vec c * vec b =0`
`therefore vec c and vec b` are perpendicular.
similarly
` (vec b xx vec c)xxvec b= vec a xx vec b `
`=>(vec b * vec b )vec c -(vec b * vec c )vec b = vec c `
`therefore vec b * vec c =0`
so, ` vec b and vec c ` are perpendicular.
again ` (vec c xx vec a)xxvec c = vec b xx vec c `
`therefore vec a * vec c =0`
`therefore vec a and vec c` are perpendicular.
hence` vec a ,\ vec b ,\ vec c`
form an orthonormal right handed triad of unit vectors