Question

If $a,c,d$ are non-coplanar vectors and $d.\{a\times [b\times (c\times d)]\}$ is equal to

• A. $(b.d)\left[acd\right]$
• B. $(a.d)\left[acd\right]$
• C. $(c.d)\left[acd\right]$
• D. None of these

Answer 1

Answer: (A)

$(b.d)\left[acd\right]$

We have $d.\{a\times [b\times (c\times d)]\}$

$=d.(a\times ((b.d)c-(b.c)d))=d.((b.d)(a\times c)-(b.c)(a\times d))=(b.d)d.(a\times c)-(b.c)d.(a\times d)=(b.d)\left[acd\right][\because d.(a\times d)=0]$