If a, b, c are three non-zero vectors, then which of the following statement(s) is/are ture?

Options

A. `veca xx (vecb xx vecc), vecb xx (vecc xx veca), (vecc xx veca), vecc xx (veca xx vecb)` form a right handed system
B. `vecc, (veca xx vecb) xx vecc, veca xx vecb` from a right handed system
C. `veca.vecb+vecb.vecc+vecc.veca lt 0` if `veca+vecb+vecc=vec0`
D. `((veca xx vecb).(vecb xx vecc))/((vecb xx vecc).(veca xx vecc))=-1` if `veca + vecb + vecc=0`

Answer

B, C, D

Explanation

a) `veca xx (vecb xx vecc) + vecb xx (vecc xx veca) + vecc xx (veca xx vecb)=vec0`
`rArr` Vectors are coplanar, so do not form RHS.
b) `(veca xx vecb) xx vecc, veca, veca xx vecb, vecc` in that order from RHS.
`rArr vecc, (veca xx vecb) xx vecc, veca xx vecb` also form RHS as they are in same cycle order.
c) `veca + vecb+vecc=0 rArr (veca +vecb+vecc)=0`
`rArr veca^(-2)+vecb^(-2)=-2a(veca.vecb+vecb.vecc+vecc.veca)`
Hence, `veca.vecb+vecb.vecc+vecc.veca lt 0`
d) `veca+vecb+vecc=0 rArr veca xx vecb=vecb xx vecc = vecc xx veca`
Using this we get result.