Prove that: `[(vecaxxvecb)xx(vecaxxvecc)].vecd=[veca vecb vecc](veca.vecd)`

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

Prove that vecaxx{vecbxx(veccxxvecd)}=(vecb.vecd)(vecaxxvecc)-(vecb.vecc)(vecaxxvecd)

Prove that: (vecaxxvecb)xx(veccxxvecd)+(vecaxxvecc)xx(vecd xx vecb)+(vecaxxvecd)xx(vecbxxvecc)=2[vecb vecc vecd] veca

for any four vectors veca,vecb, vecc and vecd prove that vecd. (vecaxx(vecbxx(veccxxvecd)))=(vecb.vecd)[veca vecc vecd]

for any four vectors veca,vecb, vecc and vecd prove that vecd. (vecaxx(vecbxx(veccxxvecd)))=(vecb.vecd)[veca vecc vecd]

If veca, vecba and vecc are non- coplanar vecotrs, then prove that |(veca.vecd)(vecbxxvecc)+(vecb.vecd)(veccxxveca)+(vecc.vecd)(vecaxxvecb) is independent of vecd where vecd is a unit vector.

If veca, vecba and vecc are non- coplanar vecotrs, then prove that |(veca.vecd)(vecbxxvecc)+(vecb.vecd)(veccxxveca)+(vecc.vecd)(vecaxxvecb) is independent of vecd where vecd is a unit vector.

If veca,vecb,vec c,vecd are the position vectors of the verticles of a cyclic quadrilateral ABCd prove that (|vecaxxvecb+vecbxxvecd+vecd xxveca|)/((vecb-veca).(vecd-veca))+(|vecbxxvec c+veccxxvecd+vecd xxvecb|)/((vecb-vecc).(vecd-vecc))=0

If veca,vecb, vecc and veca',vecb',vecc' are reciprocal system of vectors, then prove that veca'xxvecb'+vecb'xxvecc'+vecc'xxveca'=(veca+vecb+vecc)/([vecavecbvecc])

If veca,vecb,vecc and vecd are the position vectors of the vertices of a cycle quadrilateral ABCD, prove that (|vecaxxvecb+vecb xxvecd+vecd xxveca|)/((vecb-veca).(vecd-veca))+(|vecbxxvecc+veccxxvecd+vecd+vecd xx vecb|)/((vecb-vecc).(vecd-vecc))