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Question

# The vectors ¯a,¯b,¯c are coplanar, if and only if ¯a+¯b,¯b+¯c and ¯c+¯a are coplaner

A
True
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B
False
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Solution

## The correct option is A TrueThree vectors ¯¯¯x,¯¯¯yand¯¯¯z are coplanar.If:¯¯¯x(¯¯¯y×¯¯¯z)=0[xyz]=0Here, (¯¯¯a+¯¯b)[(¯¯b+¯¯c)×(¯¯c+¯¯¯a)]=0⟹(¯¯¯a+¯¯b).[(¯¯b×¯¯c)+(¯¯b×¯¯¯a)+(¯¯c×¯¯c)+(¯¯c×¯¯¯a)]=0⟹(¯¯¯a+¯¯b).[(¯¯b×¯¯c)+(¯¯b×¯¯¯a)+0+(¯¯c×¯¯¯a)]⟹¯¯¯a.(¯¯b×¯¯c)+¯¯¯a.(¯¯b×¯¯¯a)+¯¯¯a.(¯¯c×¯¯¯a)+¯¯b.(¯¯b×¯¯c)+¯¯b.(¯¯b×¯¯¯a)+¯¯b.(¯¯c×¯¯¯a)=0⟹¯¯¯a.(¯¯b×¯¯c)+0+0+0+0+¯¯b.(¯¯c×¯¯¯a)=0−(i)∵[¯¯¯a¯¯b¯¯c]=[¯¯b¯¯¯a¯¯c]⟹¯¯¯a.(¯¯b×¯¯c)=¯¯b.(¯¯¯a×¯¯c)⟹¯¯¯a.(¯¯b×¯¯c)=−¯¯b.(¯¯c×¯¯¯a)Putting in (i):¯¯¯a.(¯¯b×¯¯c)−¯¯¯a.(¯¯b×¯¯c)=0∴If¯¯¯a+¯¯b,¯¯b+¯¯cand¯¯c+¯¯¯a are coplanar, hence ¯¯¯a,¯¯band¯¯c are coplanar too.∴A)True.

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