Prove that : `veca*(vecb+vec c)xx(veca+2vecb+3vec c)=[veca vecb vec c]`

Prove that veca*(vecb+vec c)xx (veca+3vecb+2vec c)=-(veca vecb vecc )

Prove that: (2veca-vecb)xx (veca+2vecb)=5vecaxxvecb .

Prove that (veca+3vecb)xx(veca+vecb)+(3veca-5vecb)xx(veca-vecb)=0

Prove that: |(veca+vecb)xx(veca-vecb)|=2ab if veca_|_vecb

For any two vectors vec a\ a n d\ vec b prove that | vec axx vec b|^2=| (veca. veca , veca. vecb),(vecb.veca ,vecb.vecb)|

If veca , vecb and vecc are three vectors such that vecaxx vecb =vecc, vecb xx vecc= veca, vecc xx veca =vecb then prove that |veca|= |vecb|=|vecc|

If vecl,vecm,vecn are three non coplanar vectors prove that [vec vecm vecn](vecaxxvecb) =|(vec1.veca, vec1.vecb, vec1),(vecm.veca, vecm.vecb, vecm),(vecn.veca, vecn.vecb, vecn)|

If vec a , vec b , vec c are any three noncoplanar vector, then ( veca + vecb + vecc )[( veca + vecb )×( veca + vecc )] is :

i. If vec a , vec b a n d vec c are non-coplanar vectors, prove that vectors 3veca -7vecb -4 vecc ,3 veca -2vecb + vecc and veca + vecb +2 vecc are coplanar.

Prove that ( veca+ vec b).( vec a+ vecb)= | veca|^2+| vec b|^2 , if and only if vec a ,vec b are perpendicular, given vec a!= vec0, vec b!= vec0