Question

The vector $\hat{i}\times \left[\right(a\times b)\times \hat{i}]+\hat{j}\times \left[\right(a\times b)\times \hat{j}]+\hat{k}\times \left[\right(a\times b)\times \hat{k}]$ is equal to

• A. $0$
• B. $(a\cdot b)b$
• C. $b$
• D. $2(a\times b)$

Answer 1

The correct option is D $2(a\times b)$

we know that, $\overrightarrow{A}=(\overrightarrow{A}\cdot i)i+(\overrightarrow{A}\cdot j)j+(\overrightarrow{A}\cdot k)k$

$i\times ((a\times b)\times i)=(i.i)(a\times b)-(i.(a\times b))ij\times ((a\times b)\times j)=(j.j)(a\times b)-(j.(a\times b))jk\times ((a\times b)\times k)=(k.k)(a\times b)-(k.(a\times b))k$
Therefore the given expression is
$=3(a\times b)-(i.(a\times b)i+(j.a\times b)j+k\cdot (a\times b))k)=3(a\times b)-a\times b=2(a\times b)$