For any two vectors `overset(to)(u) " and " overset(to)(v)` prove that
`(i) |overset(to)(u).overset(to)(v)|^(2) +|overset(to)(u) xx overset(to)(v)|^(2) =|overset(to)(u)|^(2)| overset(to)(v)|^(2)`
`(ii) (1+|overset(to)(u)|^(2))(1+|overset(to)(v)|^(2))=|1-overset(to)(u).overset(to)(v)|^(2)+|overset(to)(u)+overset(to)(v)+(overset(to)(u)xxoverset(to)(v))|^(2)`

Explanation

(i) Since `vec(u) ". " vec(v) = |vec(u) ||vec(v)| cos0`
and `vec(u) xx vec(v) = |vec(u)||vec(v)|` sin 0 `hat(n)`
where 0 is the angle between `vec(u) " and " vec(v) " and " vec(n)` is unit vector perpendicular to the plane of `vec(u) " and " vec(v)`
Again `|ve(u) ". " vec(v)|^(2) =|vec(u)|^(2) |vec(v)|^(2) | cos^(2) 0` and
` |vec(u) xx vec(v)|^(2)= |vec(u)|^(2)|vec(v)|^(2) sin^(2) 0= |vec(u)|^(2) |vec(v)|^(2) sin^(2) 0`
`:. |vec(u)"." vec(v)|^(2) +|vec(u)xx vec(v)|^(2)=|vec(u)|^(2) |vec(v)|^(2) (cos^(2) 0+sin^(2)0)`
`=|vec(u)|^(2) |vec(v)|^(2)`
(ii) `|vec(u)+vec(v)+(vec(u)xxvec(v)|^(2)`
`|vec(u)+vec(v)|^(2) +|vec(u)xxvec(v)|^(2)+2(vec(u)+vec(v)).(vec(u)xxvec(v))`
`=|vec(u)|^(2)+|vec(v)|^(2)+2vec(u)"."vec(v)+|vec(u)xx vec(v)|^(2) +0`
[`:' vec(u) xx vec(v)` is perpendicular to the plane of `vec(u) " and " vec(v)`]
`:. |vec(u) |^(2) +|vec(v)|^(2)+2vec(u)". " vec(v) +|vec(u)xx vec(v)|^(2) +1 -2 vec(u)". " vec(v) + |vec(u) ". " vec(v)|^(2)`
`=|vec(u)|^(2) + |vec(v)|^(2) +1+ |vec(u)|^(2) |vec(v)|^(2)`
` =|vec(u)|^(2) (1+|vec(v)|^(2)) +(1+|vec(v)|^(2)) =(1+|vec(v)|^(2)) (1+|vec(u)|^(2))`