  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  B
32  C
2  D
3  Solution

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

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