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Let overset(to)(u) " and" overset(to)(v)beunitvectors. If overset(to)(w) is avector suchthatoverset(to)(w) +(overset(to)(w) xx overset(to)(u)) = overset(to)(v) Thenprovethat |(overset(to)(u) xx overset(to)( v)) . overset(to)(w) |le .(1)/(2)and that theequalityholdsif andonluy ifoverset(to)(u) is perpendicular to overset(to)(v). |
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Answer» Takingcrossproductwith `vec(u) ` we get `vec(u) xx [vec(w) + (vec(w) xx vec(u))]=vec(u) xx vec(v)` `rArrvec(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)` Nowtakigndotproductof 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)` Nowtakingdot productof Eq. (i) with ` vec(u)` we get ` vec(v) ". " vec(w)+vec(v) ". " (vec(w) xx vec(u)) =vec(v) ". " vec(v)` `rArrvec(v) ". " vec(w) + [vec(v) vec(w) vec(u)]= 1 rArrvec(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)` Takingdotproductof 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)` takingdotproductof 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 fromEq. (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 ]` Theequality holdsif andonlyif `vec(u)". " vec(v) = 0 " Iff" vec(u)` is perpendicularto `vec(v)` |
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