Question

If $\overrightarrow{a},\overrightarrow{c},\overrightarrow{d}$ are non-coplanar vectors satisfying $\overrightarrow{d}\cdot (\overrightarrow{a}\times (\overrightarrow{b}\times (\overrightarrow{c}\times \overrightarrow{d})\left)\right)=k\left[\overrightarrow{a}\overrightarrow{c}\overrightarrow{d}\right]$ and $\overrightarrow{b}\cdot \overrightarrow{d}=14$, then the value of $k$ is equal to

• A. 16
• B. 18
• C. 14
• D. 12

Answer 1

The correct option is C 14

The given expression is
$d.(a\times [(b.d)c-(b.c)d])=d.((b.d)(a\times c)-(b.c)(a\times d))=(b.d)d.(a\times c)-(b.c)d.(a\times d)=(b.d)\left[acd\right]\Rightarrow K=b.d=14$