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Let vecu and vecv be unit vectors. If ve...

Let `vecu and vecv` be unit vectors. If `vecw` is a vector such that `vecw+vecwxxvecu=vecv`, then prove that `|(vecuxxvecv).vecw|le1/2` and that the equality holds if and only if `vecu` is perpendicular to `vecv`.

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`[vecuvecv vecw] = (vecuxxvecv).(vecv-vecwxxvecu)`
`=(vecuxxvecv).(vecuxxvecw)`
`|{:(vecu.vecu,vecu.vecw),(vecv.vecu,vecv.vecw):}|`
Now `vecu.vecu=1`
`vecu.vecw=vecu.(vecv-vecw xxvecu)=vecu.vecv-[vecuvecwvecu]`
`=vecu.vecv`
`vecv.vecw=vecv.(vecv-vecwxxvecu)=1-[vecv vecw vecu]`
`=1-[vecu vecv vecw]`
`[vecuvecv vecw]=|{:(1,cos theta),(costheta,1-[vecuvecv vecw]):}|`
( `theta` is the angle between `vecu and vecv`
`1-[vecu vecv vecw] -cos^(2)theta`
`1/2 sin^(2)thetale1/2`
Equality holds when `sin^(2) theta=1, i.e., theta = pi//2, i.e., vecu bot vecv`
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