Question

The vector $i\times (a\times i)+j\times (a\times j)+k\times (a\times k)$ is equal to

• A. $0$
• B. $a$
• C. $2a$
• D. none of these

Answer 1

Answer: (C)

$2a$

$CONCEPT:$ $a=a_{1}i+a_{2}i+a_{3}k$, so that
$a.i=a_1,a\cdot j=a_2$ and $a.k=a_3$
$\Rightarrow a=(a\cdot i)i+(a\cdot j)j+(a\cdot k)k$
$i\times (a\times i)=(i\cdot i)a-(i\cdot a)i=a-(i\cdot a)i,$
$j\times (a\times j)=(j\cdot j)a-(j\cdot a)j=a-(j\cdot a)j,$
and $k\times (a\times k)=(k\cdot k)a-(k\cdot a)k=a-(k\cdot a)k,$
$\therefore i\times (a\times i)+j\times (a\times j)a+k(a\times k)$
$=3a-\left[\right(i\cdot a)i+(j\cdot a)j+(k\cdot a\left)k\right]=3a-a=2a$