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Question

Let $$a=2i+j-2k$$ and $$b=i+j$$. If $$c$$ is a vector such that $$\quad a.c=\left| c \right| .\left| c-a \right| =2\sqrt { 2 } $$ and the angle between $$\left( a\times b \right) $$ and $$c$$ is $${ 30 }^{ o }$$, then $$\left| \left( a\times b \right) \times c \right| $$ is equal to


A
23
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B
32
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C
2
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D
3
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Solution

The correct option is B $$\cfrac { 3 }{ 2 } $$
$$a\times b=\begin{vmatrix} i & j & k \\ 2 & 1 & -2 \\ 1 & 1 & 0 \end{vmatrix}=2i-2j+k\quad $$
$$\quad \therefore \left| a\times b \right| =\sqrt { 4+4+1 } =3$$
$$\left| c-a \right| =2\sqrt { 2 }$$
$$ \Rightarrow { \left| c-a \right|  }^{ 2 }$$
$$={ \left( c-a \right)  }^{ 2 }=8$$
$$\Rightarrow { \left| c \right|  }^{ 2 }-2a.c+{ \left| a \right|  }^{ 2 }=8\quad $$
$$\Rightarrow { \left| c \right|  }^{ 2 }-2\left| c \right| +9=8$$
$$\Rightarrow { \left| c \right|  }^{ 2 }-2\left| c \right| +1\Rightarrow \left| c \right| =1$$
$$\quad \therefore \left| \left( a\times b \right) \times c \right| =\left|( a\times b)c \right| \sin { { 30 }^{ o } } =\cfrac { 3 }{ 2 } $$

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