Show that vectors `bara times (barb times barc),barb times (barc times bara). barc times (bara times barb)` are coplanar.

bara times (barb times barc) +barb times (barc times bara)+ barc times (bara times barb) equals

IF bara times barb = barc and barb times barc=bara then

If barc=bara times barb and barb=barc times bara then

IF barx times barb=barc times barb and barx . bara =0 then barx=

Let barlamda=bara times (barb +barc), barmu=barb times (barc+bara) and barv=barc times (bara+barb) , Then

IF OABC is a tetrahedron where O is the origin and A,B and C have respective positions vectors as bara,barb and barc then prove that the circumcentre of the tetrahedron is ((bara)^2(barb times barc)+(barb)^2 (barc times bara)+(barc)^2 (bara times barb))/(2[bara barb barc])

If hatd is a unit vectors such that hatd=lamdabarb times barc+mubarc times bara+vbara times barb then |(hatd.bara)(barb times barc)+(bard.barb) (barc times bara)+(bard.barc) (bara times barc)| is equal to

bara,barb,barc,bard are four distinct vectors satisfying the conditions bar a times barb =barc times bard and bara times barc=barb times bard then show that bara.barb+bard.barc ne bard.barb+bara.barc.