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If $$A=2i+k,B=i+j+k$$ and $$C=4i-3j+7k$$, then a vector $$r$$ which satisfies $$R\times B=C\times B$$ and $$R.A=0$$, is


A
i8j+2k
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B
i8j+2k
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C
i+8j+2k
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D
none of these
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Solution

The correct option is A $$-i-8j+2k$$
Let $$R=xi+yj+zk$$
$$\therefore R.A=0\Rightarrow 2x+z=0$$   ....(1)
$$R\times B=C\times B\Rightarrow \begin{vmatrix} i & j & k \\ x & y & z \\ 1 & 1 & 1 \end{vmatrix}=\begin{vmatrix} i & j & k \\ 4 & -3 & 7 \\ 1 & 1 & 1 \end{vmatrix}\\ \Rightarrow \left( y-z \right) i+\left( z-x \right) j+\left( x-y \right) k=-10i+3j+7k$$
$$\Rightarrow y-z=-10$$   ...(2)
$$z-x=3$$    ...(3)
and $$x-y=7$$
Solving (1) and (2), we get $$x=-1,z=2$$
$$\therefore$$ From (2), $$y=-8$$
Hence $$R=-i-8j+2k$$

Mathematics

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