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Question

The position vectors of four points P,Q,R,S are 2¯¯¯a+4¯¯c,5¯¯¯a+33¯¯b+4¯¯c),23¯¯b+¯¯c and 2¯¯¯a+¯¯c respectively, then

A
¯¯¯¯¯¯¯¯PQ is parallel to ¯¯¯¯¯¯¯¯RS
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B
¯¯¯¯¯¯¯¯PQ is not parallel to ¯¯¯¯¯¯¯¯RS
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C
¯¯¯¯¯¯¯¯PQ is equal to ¯¯¯¯¯¯¯¯RS
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D
¯¯¯¯¯¯¯¯PQ is parallel and equal to ¯¯¯¯¯¯¯¯RS
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Solution

The correct option is A ¯¯¯¯¯¯¯¯PQ is parallel to ¯¯¯¯¯¯¯¯RS
P(2a+4c) Q(5a+33b+4c)

R(23b+c) S(2a+c)

PQ=OQOP=(5a+33b+4¯c)(2a+4c)

PQ=(3¯a+33¯b)=3(a+3b)

RS=OSOR=(2¯a+¯c)(23b+c)

RS=(2a+23¯b)=2(a+3b)

PQ=3(a+3b)RS=(¯a+3b)

Since directions of PQ and RS are same

PQ is parallel to RS

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