Question

If $ {a}=(1,2,3), {b}=(2,-1,1),{c}=(3,2,1) $ and $ {a} \times ( {b}\times {c} )= \alpha \,{a}+ \beta\,{b}\times \gamma\, {c} $ then

• A. $\alpha=1,\beta=10,\gamma=3$
• B. $\alpha=0,\beta=10,\gamma=-3$
• C. $\alpha+\beta+\gamma=8$
• D. $\alpha = \beta = \gamma = 0$

Answer 1

Answer: (B)

Given,
$a =(1,2,3)= i +2 j +3 k $
$b =(2,-1,1)=2 i - j + k$
and $c=(3,2,1)=3 i+2 j+k$
Now, $ a \times( b \times c )=\alpha a +\beta b +\gamma c $
$\Rightarrow ( a \cdot c ) b -( a \cdot b ) c =\alpha a +\beta b +\gamma c $
$\Rightarrow \{( i +2 j +3 k ) \cdot(3 i +2 j + k )\} b $
$ -\{( i +2 j +3 k ) \cdot( 2 i - j + k )\} c $
$ =\alpha a +\beta b +\gamma c$
$\Rightarrow (3+4+3) b -(2-2+3) c =\alpha a +\beta b +\gamma c $
$\Rightarrow 0 a +10 b-3 c=\alpha a +\beta b +\gamma c $
On comparing, we get
$\alpha=0, \beta=10 \text { and } \gamma=-3$