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Let overset(to)(u) " and " overset(to)(...

Let `overset(to)(u) " and " overset(to)(v) ` be unit vectors . If `overset(to)(w)` is a vector such that `overset(to)(w) + (overset(to)(w) xx overset(to)(u)) = overset(to)(v)` Then prove that
`|(overset(to)(u) xx overset(to)( v)) . overset(to)(w) | le .(1)/(2) ` and that the equality holds if and onluy if `overset(to)(u)` is perpendicular to `overset(to)(v)`.

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Given equation is `vec(w) +v(vec(w) xx vec(u)) = vec(v)`
Taking cross product with `vec(u) ` we get
`vec(u) xx [vec(w) + (vec(w) xx vec(u))]=vec(u) xx vec(v)`
`rArr vec(u) xx vec(w) + vec(u) (vec(w) xx vec(u)) = vec(u) xx vec(v)`
`rArr vec(u) xx vec(w) + (vec(u) "." vec(u)) vec(w) - (vec(u)"." vec(w)) =vec(u) =vec(u) xx vec(v)`
Now takign dot product of Eq. (i) with `vec(u)` we get
` vec(u) "." vec(w) +vec(u) "."(vec(w) xx vec(u)) =vec(u) "." vec(v)`
`rArr vec(u) ". " vec(w) = vec(u) ". " vec(v) [ :' vec(u) . (vec(w) xx vec(u)) =vec(v) "." vec(v)`
Now taking dot product of Eq. (i) with ` vec(u)` we get
` vec(v) ". " vec(w) +vec(v) ". " (vec(w) xx vec(u)) =vec(v) ". " vec(v)`
`rArr vec(v) ". " vec(w) + [vec(v) vec(w) vec(u)]= 1 rArr vec(v) ". " vec(w) + [ vec(v) vec(w) vec(u)] -1=0`
`rArr -(vec(u) xx vec(u)) ". " vec(w) - vec(v) ". " vec(w) +1=0`
`rArr 1-vec(v) ". " vec(w) = (vec(u) xx vec(v)) ". " vec(w)`
Taking dot product of Eq (ii) with `vec(w)` we get
`(vec(u)xx vec(w)) ". " vec(w) + vec(w) ". " vec(w)-vec(u)"." vec(w)) (vec(u)"." vec(w)) =(vec(u)xxvec(v))" ."vec(w)`
`rArr 0+ |vec(w)|^(2) -(vec(u)". " vec(w))^(2) =(vec(u) xx vec(v))"." vec(w)`
`rArr (vec(u) xx vec(v)) ". " vec(w) =|vec(w)|^(2)-(vec(u) ". " vec(w))^(2)`
taking dot product of Eq. (i) with `vec(w)` we get
`vec(w) ". " vec(w) + (vec(w) xx vec(u)) ". " vec(w) = vec(v) ". " vec(w)`
`rArr |vec(w)|^(2) =1-(vec(u) xx vec(v))"." vec(w)`
Again from Eq. (v) we get
`(vec(u) xx vec(v)) "."vec(w)|vec(w)|^(2) - (vec(u)"." vec(w))^(2) =1- (vec(u) xx vec(v)) "." vec(w) - (vec(u)"."vec(w))^(2)`
`rArr 2(vec(u) ". " vec(v)) "." vec(w) =1 - (vec(u)"." vec(v))^(2)`
`rArr |(vec(u) xx vec(w)) "." vec(w)|=(1)/(2)|1-(vec(u) "." vec(v))^(2) |le (1)/(2) [ :' (vec(u) "." vec(v))^(2) ge 0 ]`
The equality holds if and only if `vec(u) ". " vec(v) = 0 " Iff " vec(u)` is perpendicular to `vec(v)`
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