If →a,→b,→care three non-coplanar vectors,→p,→q,→rare non-zero vectors such that→p=→b×→c[→a→b→c],→q=→c×→a[→a→b→c],→r=→a×→b[→a→b→c], then the value of the expression (→a+→b+→c),(→p+→q+→r) is
A
0
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B
4
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C
2
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D
3
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Solution
The correct option is A0 [¯¯¯a¯¯b¯¯c]≠0¯¯¯¯P=¯bׯc[¯¯¯a¯b¯c],¯¯¯¯Q=¯cׯ¯¯a[¯¯¯a¯b¯¯¯c]¯¯¯¯R=¯¯¯aׯb[¯¯¯a¯b¯c]Tofind(¯¯¯a+¯¯b+¯¯c).(¯¯¯p+¯¯¯q+¯¯¯r)¯¯¯a.¯¯¯p=¯¯¯a.(¯bׯc)[¯¯¯a¯b¯¯¯c]=1[¯¯¯a¯b¯¯¯c]={¯¯¯a.(¯¯bׯ¯c)}=[¯¯¯a¯b¯c][¯¯¯a¯b¯c]=1¯¯b.¯¯¯p=¯¯b.(¯bׯc)[¯¯¯a¯b¯¯¯c]=1[¯¯¯a¯b¯¯¯c]{¯¯b.(¯¯bׯ¯c)}=0¯¯c.¯¯¯p=0¯¯b.¯¯¯q=1¯¯c.¯¯¯r=1RemainingZero.