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If veca,vecb,vecc are non-coplanar vecto...

If `veca,vecb,vecc` are non-coplanar vectors and `vecu` and `vecv` are any two vectors. Prove that `vecuxxvecv=(1)/([vecavecbvecc])|{:(vecu.veca,vecv.veca,veca),(vecu.vecb,vecv.vecb,vecb),(vecu.vecc,vecv.vecc,vecc):}|`

A

`|vec u|+ vecu. (veca x vecb)`

B

`|vecu|+ |vecu. veca|`

C

`|vecu| + |vecu.vecb|`

D

`|vecu|+ vecu. (veca + vecb)`

Text Solution

Verified by Experts

The correct Answer is:
a,c
we have
`vecv= vecaxxvecb= |veca||vecb| sin theta hatn = sin theta hatn`
where `veca and vecb` are unit vectors. Therefore,
`|vecv|= sin theta`
Now, `vecu = veca - (veca.vecb)vecb`
`= veca -vecb cos theta ( " where " veca. Vecb = cos theta)`
`|vecu|^(2) = | veca-vecb cos theta|^(2)`
` 1 + cos^(2) theta -2 cos theta . cos theta`
` =1 - cos^(2) theta = sin^(2) theta = |v|^(2)`
` Rightarrow |vecu|= |vecv|`
Also , `vecu . vecb = veca. vecb - (veca.vecb) (vecb.vecb)`
` = veca.vecb-veca.vecb=0`
`|vecu.vecb|=0`
`|vecv|=|vecu|+ |vecu.vecb|` is also correct.
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